Use a Problem Solving Strategy

Use a Problem Solving Strategy for Word Problems

Now that we can solve equations, we are ready to apply our new skills to word problems. We will develop a strategy we can use to solve any word problem successfully.

Example

Try it.

Normal yearly snowfall at the local ski resort is 12 inches more than twice the amount it received last season. The normal yearly snowfall is 62 inches. What was the snowfall last season at the ski resort?

Solution

Step 1. Read the problem.
Step 2. Identify what you are looking for.	What was the snowfall last season?
Step 3. Name what we are looking for and choose a variable to represent it.	Let \(s=\) the snowfall last season.
Step 4. Translate. Restate the problem in one sentence with all the important information.
Translate into an equation.
Step 5. Solve the equation.
Subtract 12 from each side.
Simplify.
Divide each side by two.
Simplify.
Step 6. Check: First, is our answer reasonable? Yes, having 25 inches of snow seems OK. The problem says the normal snowfall is twelve inches more than twice the number of last season. Twice 25 is 50 and 12 more than that is 62.
Step 7. Answer the question.	The snowfall last season was 25 inches.

We summarize an effective strategy for problem solving.

Solve Number Word Problems

We will now apply the problem solving strategy to “number word problems.” Number word problems give some clues about one or more numbers and we use these clues to write an equation. Number word problems provide good practice for using the Problem Solving Strategy.

Example

Try it.

The sum of seven times a number and eight is thirty-six. Find the number.

Solution

Step 1. Read the problem.
Step 2. Identify what you are looking for.	the number
Step 3. Name what you are looking for and choose a variable to represent it.	Let n = the number.
Step 4. Translate: Restate the problem as one sentence. Translate into an equation.
Step 5. Solve the equation. Subtract eight from each side and simplify. Divide each side by seven and simplify.
Step 6. Check. Is the sum of seven times four plus eight equal to 36? \(\ \begin{array}{lll}7\cdot 4+8 & \overset{?}{=} & 36 \\ 28+8 & \overset{?}{=} & 36 \\ 36 & = & 36✓\end{array}\)
Step 7. Answer the question.	The number is 4.

Did you notice that we left out some of the steps as we solved this equation? If you’re not yet ready to leave out these steps, write down as many as you need.

Some number word problems ask us to find two or more numbers. It may be tempting to name them all with different variables, but so far, we have only solved equations with one variable. In order to avoid using more than one variable, we will define the numbers in terms of the same variable. Be sure to read the problem carefully to discover how all the numbers relate to each other.

Example

Try it.

The sum of two numbers is negative fifteen. One number is nine less than the other. Find the numbers.

Solution

Step 1. Read the problem.
Step 2. Identify what you are looking for.	two numbers
Step 3. Name what you are looking for by choosing a variable to represent the first number. “One number is nine less than the other.”	Let \(n={1}^{\text{st}}\) number. \(n-9={2}^{\text{nd}}\) number
Step 4. Translate. Write as one sentence. Translate into an equation.	The sum of two numbers is negative fifteen.
Step 5. Solve the equation. Combine like terms. Add nine to each side and simplify. Simplify.
Step 6. Check. Is \(-12\) nine less than \(-3?\) \(\ \begin{array}{lll}-3-9 & \overset{?}{=} & -12 \\ -12 & = & -12✓\end{array}\) Is their sum \(-15?\) \(\ \begin{array}{lll}-3+(-12) & \overset{?}{=} & -15 \\ -15 & = & -15✓\end{array}\)
Step 7. Answer the question.	The numbers are \(-3\) and \(-12.\)

We will use this notation to represent consecutive integers in the next example.

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Solve Percent Applications

There are several methods to solve percent equations. In algebra, it is easiest if we just translate English sentences into algebraic equations and then solve the equations. Be sure to change the given percent to a decimal before you use it in the equation.

Example

Try it.

Translate and solve:

ⓐ What number is 45% of 84?ⓑ 8.5% of what amount is $4.76? ⓒ 168 is what percent of 112?

Solution

ⓐ

Translate into algebra. Let n = the number.
Multiply.
			37.8 is 45% of 84.

ⓑ

Translate. Let n = the amount.
Multiply.
Divide both sides by 0.085 and simplify.
			8.5% of $56 is $4.76

ⓒ

We are asked to find percent, so we must have our result in percent form.
Translate into algebra. Let p = the percent.
Multiply.
Divide both sides by 112 and simplify.
Convert to percent.
			168 is 150% of 112.

Now that we have a problem solving strategy to refer to, and have practiced solving basic percent equations, we are ready to solve percent applications. Be sure to ask yourself if your final answer makes sense—since many of the applications we will solve involve everyday situations, you can rely on your own experience.

Example

Try it.

The label on Audrey’s yogurt said that one serving provided 12 grams of protein, which is 24% of the recommended daily amount. What is the total recommended daily amount of protein?

Solution

What are you asked to find?	What total amount of protein is recommended?
Choose a variable to represent it.	Let \(a=\) total amount of protein.
Write a sentence that gives the information to find it.
Translate into an equation.
Solve.
Check: Does this make sense? Yes, 24% is about \(\frac{1}{4}\) of the total and 12 is about \(\frac{1}{4}\) of 50.
Write a complete sentence to answer the question.	The amount of protein that is recommended is 50 g.

Remember to put the answer in the form requested. In the next example we are looking for the percent.

Applications of discount and mark-up are very common in retail settings.

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Solve Simple Interest Applications

Interest is a part of our daily lives. From the interest earned on our savings to the interest we pay on a car loan or credit card debt, we all have some experience with interest in our lives.

The amount of money you initially deposit into a bank is called the principal, P, and the bank pays you interest, I. When you take out a loan, you pay interest on the amount you borrow, also called the principal.

In either case, the interest is computed as a certain percent of the principal, called the rate of interest, r. The rate of interest is usually expressed as a percent per year, and is calculated by using the decimal equivalent of the percent. The variable t, (for time) represents the number of years the money is saved or borrowed.

Interest is calculated as simple interest or compound interest. Here we will use simple interest.

The formula we use to calculate interest is \(I=Prt.\) To use the formula we substitute in the values for variables that are given, and then solve for the unknown variable. It may be helpful to organize the information in a chart.

Example

Try it.

Areli invested a principal of $950 in her bank account that earned simple interest at an interest rate of 3%. How much interest did she earn in five years?

Solution

\(\begin{array}{lll}I & = & ? \\ P & = & \text{\$}950 \\ r & = & 3\% \\ t & = & 5\ \text{years}\end{array}\)

Identify what you are asked to find, and choose a variable to represent it.	What is the simple interest? \(\text{Let}\ I=\text{interest.}\)
Write the formula.	\(I=Prt\)
Substitute in the given information.	\(I=(950)(0.03)(5)\)
Simplify.	\(I=142.5\)
Check.
Is $142.50 a reasonable amount of interest on $950?
Yes.
Write a complete sentence.	The interest is $142.50.

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Key Concepts

• How To Use a Problem Solving Strategy for Word Problems
  1. Read the problem. Make sure all the words and ideas are understood.
  2. Identify what you are looking for.
  3. Name what you are looking for. Choose a variable to represent that quantity.
  4. Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebra equation.
  5. Solve the equation using proper algebra techniques.
     

     
     
     

  6. Check the answer in the problem to make sure it makes sense.
  7. Answer the question with a complete sentence.
• How To Find Percent Change
  1. Find the amount of change
     \(\text{change}=\text{new amount}-\text{original amount}\)
  2. Find what percent the amount of change is of the original amount.
     \(\text{change is what percent of the original amount?}\)
• Discount
  \(\begin{array}{lll}\text{amount of discount} & = & \text{discount rate}\cdot \text{original price} \\ \\ \text{sale price} & = & \text{original amount}-\text{discount}\end{array}\)
• Mark-up
  \(\begin{array}{lll}\text{amount of mark-up} & = & \text{mark-up rate}\cdot \text{original cost} \\ \\ \text{list price} & = & \text{original cost}\ +\ \text{mark up}\end{array}\)
• Simple Interest
  If an amount of money, P, called the principal, is invested or borrowed for a period of t years at an annual interest rate r, the amount of interest, I, earned or paid is:
  \[\begin{array}{lllll} & & I & = & \text{interest} \\ I=Prt & \ \text{where}\ & P & = & \text{principal} \\ & & r & = & \text{rate} \\ & & t & = & \text{time}\end{array}\]

Use a Problem Solving Strategy

Use a Problem Solving Strategy for Word Problems

Try it.

List five positive thoughts you can say to yourself that will help you approach word problems with a positive attitude. You may want to copy them on a sheet of paper and put it in the front of your notebook, where you can read them often.

Solution

Answers will vary.

Try it.

List five negative thoughts that you have said to yourself in the past that will hinder your progress on word problems. You may want to write each one on a small piece of paper and rip it up to symbolically destroy the negative thoughts.

In the following exercises, solve using the problem solving strategy for word problems. Remember to write a complete sentence to answer each question.

Try it.

There are 16 girls in a school club. The number of girls is four more than twice the number of boys. Find the number of boys.

Solution

six boys

Try it.

There are 18 Cub Scouts in Troop 645. The number of scouts is three more than five times the number of adult leaders. Find the number of adult leaders.

Try it.

Huong is organizing paperback and hardback books for her club’s used book sale. The number of paperbacks is 12 less than three times the number of hardbacks. Huong had 162 paperbacks. How many hardback books were there?

Solution

58 hardback books

Try it.

Jeff is lining up children’s and adult bicycles at the bike shop where he works. The number of children’s bicycles is nine less than three times the number of adult bicycles. There are 42 adult bicycles. How many children’s bicycles are there?

Solve Number Word Problems

In the following exercises, solve each number word problem.

Try it.

The difference of a number and 12 is three. Find the number.

Solution

15

Try it.

The difference of a number and eight is four. Find the number.

Try it.

The sum of three times a number and eight is 23. Find the number.

Solution

5

Try it.

The sum of twice a number and six is 14. Find the number.

Try it.

The difference of twice a number and seven is 17. Find the number.

Solution

12

Try it.

The difference of four times a number and seven is 21. Find the number.

Try it.

Three times the sum of a number and nine is 12. Find the number.

Solution

\(-5\)

Try it.

Six times the sum of a number and eight is 30. Find the number.

Try it.

One number is six more than the other. Their sum is 42. Find the numbers.

Solution

18, 24

Try it.

One number is five more than the other. Their sum is 33. Find the numbers.

Try it.

The sum of two numbers is 20. One number is four less than the other. Find the numbers.

Solution

8, 12

Try it.

The sum of two numbers is 27. One number is seven less than the other. Find the numbers.

Try it.

One number is 14 less than another. If their sum is increased by seven, the result is 85. Find the numbers.

Solution

32, 46

Try it.

One number is 11 less than another. If their sum is increased by eight, the result is 71. Find the numbers.

Try it.

The sum of two numbers is 14. One number is two less than three times the other. Find the numbers.

Solution

4, 10

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The sum of two numbers is zero. One number is nine less than twice the other. Find the numbers.

Try it.

The sum of two consecutive integers is 77. Find the integers.

Solution

38, 39

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The sum of two consecutive integers is 89. Find the integers.

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The sum of three consecutive integers is 78. Find the integers.

Solution

25, 26, 27

Try it.

The sum of three consecutive integers is 60. Find the integers.

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Find three consecutive integers whose sum is \(-36.\)

Solution

\(-11,-12,-13\)

Try it.

Find three consecutive integers whose sum is \(-3.\)

Try it.

Find three consecutive even integers whose sum is 258.

Solution

84, 86, 88

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Find three consecutive even integers whose sum is 222.

Try it.

Find three consecutive odd integers whose sum is \(-213.\)

Solution

\(-69,-71,-73\)

Try it.

Find three consecutive odd integers whose sum is \(-267.\)

Try it.

Philip pays $1,620 in rent every month. This amount is $120 more than twice what his brother Paul pays for rent. How much does Paul pay for rent?

Solution

$750

Try it.

Marc just bought an SUV for $54,000. This is $7,400 less than twice what his wife paid for her car last year. How much did his wife pay for her car?

Try it.

Laurie has $46,000 invested in stocks and bonds. The amount invested in stocks is $8,000 less than three times the amount invested in bonds. How much does Laurie have invested in bonds?

Solution

$13,500

Try it.

Erica earned a total of $50,450 last year from her two jobs. The amount she earned from her job at the store was $1,250 more than three times the amount she earned from her job at the college. How much did she earn from her job at the college?

Solve Percent Applications

In the following exercises, translate and solve.

Try it.

ⓐ What number is 45% of 120? ⓑ 81 is 75% of what number? ⓐ What percent of 260 is 78?

Solution

ⓐ 54 ⓑ 108 ⓐ 30%

Try it.

ⓐ What number is 65% of 100? ⓑ 93 is 75% of what number? ⓐ What percent of 215 is 86?

Try it.

ⓐ 250% of 65 is what number? ⓑ 8.2% of what amount is $2.87? ⓐ 30 is what percent of 20?

Solution

ⓐ \(162.5\) ⓑ $35 ⓐ 150%

Try it.

ⓐ 150% of 90 is what number? ⓑ 6.4% of what amount is $2.88? ⓐ 50 is what percent of 40?

In the following exercises, solve.

Try it.

Geneva treated her parents to dinner at their favorite restaurant. The bill was $74.25. Geneva wants to leave 16% of the total bill as a tip. How much should the tip be?

Solution

\(\text{\$}11.88\)

Try it.

When Hiro and his co-workers had lunch at a restaurant near their work, the bill was $90.50. They want to leave 18% of the total bill as a tip. How much should the tip be?

Try it.

One serving of oatmeal has 8 grams of fiber, which is 33% of the recommended daily amount. What is the total recommended daily amount of fiber?

Solution

24.2 g

Try it.

One serving of trail mix has 67 grams of carbohydrates, which is 22% of the recommended daily amount. What is the total recommended daily amount of carbohydrates?

Try it.

A bacon cheeseburger at a popular fast food restaurant contains 2070 milligrams (mg) of sodium, which is 86% of the recommended daily amount. What is the total recommended daily amount of sodium?

Solution

2407 mg

Try it.

A grilled chicken salad at a popular fast food restaurant contains 650 milligrams (mg) of sodium, which is 27% of the recommended daily amount. What is the total recommended daily amount of sodium?

Try it.

The nutrition fact sheet at a fast food restaurant says the fish sandwich has 380 calories, and 171 calories are from fat. What percent of the total calories is from fat?

Solution

45%

Try it.

The nutrition fact sheet at a fast food restaurant says a small portion of chicken nuggets has 190 calories, and 114 calories are from fat. What percent of the total calories is from fat?

Try it.

Emma gets paid $3,000 per month. She pays $750 a month for rent. What percent of her monthly pay goes to rent?

Solution

25%

Try it.

Dimple gets paid $3,200 per month. She pays $960 a month for rent. What percent of her monthly pay goes to rent?

In the following exercises, solve.

Try it.

Tamanika received a raise in her hourly pay, from $15.50 to $17.36. Find the percent change.

Solution

12%

Try it.

Ayodele received a raise in her hourly pay, from $24.50 to $25.48. Find the percent change.

Try it.

Annual student fees at the University of California rose from about $4,000 in 2000 to about $12,000 in 2010. Find the percent change.

Solution

200%

Try it.

The price of a share of one stock rose from $12.50 to $50. Find the percent change.

Try it.

A grocery store reduced the price of a loaf of bread from $2.80 to $2.73. Find the percent change.

Solution

\(-2.5\%\)

Try it.

The price of a share of one stock fell from $8.75 to $8.54. Find the percent change.

Try it.

Hernando’s salary was $49,500 last year. This year his salary was cut to $44,055. Find the percent change.

Solution

\(-11\%\)

Try it.

In ten years, the population of Detroit fell from 950,000 to about 712,500. Find the percent change.

In the following exercises, find ⓐ the amount of discount and ⓑ the sale price.

Try it.

Janelle bought a beach chair on sale at 60% off. The original price was $44.95.

Solution

ⓐ \(\text{\$}26.97\) ⓑ \(\text{\$}17.98\)

Try it.

Errol bought a skateboard helmet on sale at 40% off. The original price was $49.95.

In the following exercises, find ⓐ the amount of discount and ⓑ the discount rate (Round to the nearest tenth of a percent if needed.)

Try it.

Larry and Donna bought a sofa at the sale price of $1,344. The original price of the sofa was $1,920.

Solution

ⓐ $576 ⓑ 30%

Try it.

Hiroshi bought a lawnmower at the sale price of $240. The original price of the lawnmower is $300.

In the following exercises, find ⓐ the amount of the mark-up and ⓑ the list price.

Try it.

Daria bought a bracelet at original cost $16 to sell in her handicraft store. She marked the price up 45%. What was the list price of the bracelet?

Solution

ⓐ \(\text{\$}7.20\) ⓑ \(\text{\$}23.20\)

Try it.

Regina bought a handmade quilt at original cost $120 to sell in her quilt store. She marked the price up 55%. What was the list price of the quilt?

Try it.

Tom paid $0.60 a pound for tomatoes to sell at his produce store. He added a 33% mark-up. What price did he charge his customers for the tomatoes?

Solution

ⓐ \(\text{\$}0.20\) ⓑ \(\text{\$}0.80\)

Try it.

Flora paid her supplier $0.74 a stem for roses to sell at her flower shop. She added an 85% mark-up. What price did she charge her customers for the roses?

Solve Simple Interest Applications

In the following exercises, solve.

Try it.

Casey deposited $1,450 in a bank account that earned simple interest at an interest rate of 4%. How much interest was earned in two years?

Solution

$116

Try it.

Terrence deposited $5,720 in a bank account that earned simple interest at an interest rate of 6%. How much interest was earned in four years?

Try it.

Robin deposited $31,000 in a bank account that earned simple interest at an interest rate of 5.2%. How much interest was earned in three years?

Solution

$4836

Try it.

Carleen deposited $16,400 in a bank account that earned simple interest at an interest rate of 3.9% How much interest was earned in eight years?

Try it.

Hilaria borrowed $8,000 from her grandfather to pay for college. Five years later, she paid him back the $8,000, plus $1,200 interest. What was the rate of simple interest?

Solution

3%

Try it.

Kenneth lent his niece $1,200 to buy a computer. Two years later, she paid him back the $1,200, plus $96 interest. What was the rate of simple interest?

Try it.

Lebron lent his daughter $20,000 to help her buy a condominium. When she sold the condominium four years later, she paid him the $20,000, plus $3,000 interest. What was the rate of simple interest?

Solution

\(3.75\%\)

Try it.

Pablo borrowed $50,000 to start a business. Three years later, he repaid the $50,000, plus $9,375 interest. What was the rate of simple interest?

Try it.

In 10 years, a bank account that paid 5.25% simple interest earned $18,375 interest. What was the principal of the account?

Solution

$35,000

Try it.

In 25 years, a bond that paid 4.75% simple interest earned $2,375 interest. What was the principal of the bond?

Try it.

Joshua’s computer loan statement said he would pay $1,244.34 in simple interest for a three-year loan at 12.4%. How much did Joshua borrow to buy the computer?

Solution

$3345

Try it.

Margaret’s car loan statement said she would pay $7,683.20 in simple interest for a five-year loan at 9.8%. How much did Margaret borrow to buy the car?

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Approach Word Problems with a Positive Attitude

“If you think you can… or think you can’t… you’re right.”—Henry Ford

The world is full of word problems! Will my income qualify me to rent that apartment? How much punch do I need to make for the party? What size diamond can I afford to buy my girlfriend? Should I fly or drive to my family reunion?

How much money do I need to fill the car with gas? How much tip should I leave at a restaurant? How many socks should I pack for vacation? What size turkey do I need to buy for Thanksgiving dinner, and then what time do I need to put it in the oven? If my sister and I buy our mother a present, how much does each of us pay?

Now that we can solve equations, we are ready to apply our new skills to word problems. Do you know anyone who has had negative experiences in the past with word problems? Have you ever had thoughts like the student below?

When we feel we have no control, and continue repeating negative thoughts, we set up barriers to success. We need to calm our fears and change our negative feelings.

Start with a fresh slate and begin to think positive thoughts. If we take control and believe we can be successful, we will be able to master word problems! Read the positive thoughts in and say them out loud.

Think of something, outside of school, that you can do now but couldn’t do 3 years ago. Is it driving a car? Snowboarding? Cooking a gourmet meal? Speaking a new language? Your past experiences with word problems happened when you were younger—now you’re older and ready to succeed!

Use a Problem-Solving Strategy for Word Problems

We have reviewed translating English phrases into algebraic expressions, using some basic mathematical vocabulary and symbols. We have also translated English sentences into algebraic equations and solved some word problems. The word problems applied math to everyday situations. We restated the situation in one sentence, assigned a variable, and then wrote an equation to solve the problem. This method works as long as the situation is familiar and the math is not too complicated.

Now, we’ll expand our strategy so we can use it to successfully solve any word problem. We’ll list the strategy here, and then we’ll use it to solve some problems. We summarize below an effective strategy for problem solving.

Let’s try this approach with another example.

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Solve Number Problems

Now that we have a problem solving strategy, we will use it on several different types of word problems. The first type we will work on is “number problems.” Number problems give some clues about one or more numbers. We use these clues to write an equation. Number problems don’t usually arise on an everyday basis, but they provide a good introduction to practicing the problem solving strategy outlined above.

Example

Try it.

The difference of a number and six is 13. Find the number.

Solution

Step 1. Read the problem. Are all the words familiar?
Step 2. Identify what we are looking for.	the number
Step 3. Name. Choose a variable to represent the number.	Let \(n=\) the number.
Step 4. Translate. Remember to look for clue words like "difference... of... and..."
Restate the problem as one sentence.
Translate into an equation.
Step 5. Solve the equation.
Simplify.
Step 6. Check.
The difference of 19 and 6 is 13. It checks!
Step 7. Answer the question.	The number is 19.

Example

Try it.

The sum of twice a number and seven is 15. Find the number.

Solution

Step 1. Read the problem.
Step 2. Identify what we are looking for.	the number
Step 3. Name. Choose a variable to represent the number.	Let \(n=\) the number.
Step 4. Translate.
Restate the problem as one sentence.
Translate into an equation.
Step 5. Solve the equation.
Subtract 7 from each side and simplify.
Divide each side by 2 and simplify.
Step 6. Check.
Is the sum of twice 4 and 7 equal to 15?
\(\begin{array}{lll}2⋅4+7 & ≟ & 15 \\ 15 & = & 15✓\end{array}\)
Step 7. Answer the question.	The number is 4.

Did you notice that we left out some of the steps as we solved this equation? If you’re not yet ready to leave out these steps, write down as many as you need.

Some number word problems ask us to find two or more numbers. It may be tempting to name them all with different variables, but so far we have only solved equations with one variable. In order to avoid using more than one variable, we will define the numbers in terms of the same variable. Be sure to read the problem carefully to discover how all the numbers relate to each other.

\[\begin{array}{l}1,2,3,4 \\ \\ -10,-9,-8,-7 \\ 150,151,152,153\end{array}\]

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Key Concepts

• Problem-Solving Strategy
  1. Read the problem. Make sure all the words and ideas are understood.
  2. Identify what we are looking for.
  3. Name what we are looking for. Choose a variable to represent that quantity.
  4. Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebra equation.
  5. Solve the equation using good algebra techniques.
  6. Check the answer in the problem and make sure it makes sense.
  7. Answer the question with a complete sentence.
• Consecutive Integers
  Consecutive integers are integers that immediately follow each other.
  \[\begin{array}{llll}n & & & {1}^{\text{st}}\ \text{integer} \\ n+1 & & & {2}^{\text{nd}}\ \text{integer consecutive integer} \\ n+2 & & & {3}^{\text{rd}}\ \text{consecutive integer . . . etc.}\end{array}\]
  Consecutive even integers are even integers that immediately follow one another.
  \[\begin{array}{llll}n & & & {1}^{\text{st}}\ \text{integer} \\ n+2 & & & {2}^{\text{nd}}\ \text{integer consecutive integer} \\ n+4 & & & {3}^{\text{rd}}\ \text{consecutive integer . . . etc.}\end{array}\]
  Consecutive odd integers are odd integers that immediately follow one another.
  \[\begin{array}{llll}n & & & {1}^{\text{st}}\ \text{integer} \\ n+2 & & & {2}^{\text{nd}}\ \text{integer consecutive integer} \\ n+4 & & & {3}^{\text{rd}}\ \text{consecutive integer . . . etc.}\end{array}\]

Use a Problem-Solving Strategy

Use the Approach Word Problems with a Positive Attitude

In the following exercises, prepare the lists described.

Try it.

List five positive thoughts you can say to yourself that will help you approach word problems with a positive attitude. You may want to copy them on a sheet of paper and put it in the front of your notebook, where you can read them often.

Solution

Answers will vary

Try it.

List five negative thoughts that you have said to yourself in the past that will hinder your progress on word problems. You may want to write each one on a small piece of paper and rip it up to symbolically destroy the negative thoughts.

Use a Problem-Solving Strategy for Word Problems

In the following exercises, solve using the problem solving strategy for word problems. Remember to write a complete sentence to answer each question.

Try it.

Two-thirds of the children in the fourth-grade class are girls. If there are 20 girls, what is the total number of children in the class?

Solution

30

Try it.

Three-fifths of the members of the school choir are women. If there are 24 women, what is the total number of choir members?

Try it.

Zachary has 25 country music CDs, which is one-fifth of his CD collection. How many CDs does Zachary have?

Solution

125

Try it.

One-fourth of the candies in a bag of M&M’s are red. If there are 23 red candies, how many candies are in the bag?

Try it.

There are 16 girls in a school club. The number of girls is four more than twice the number of boys. Find the number of boys.

Solution

6

Try it.

There are 18 Cub Scouts in Pack 645. The number of scouts is three more than five times the number of adult leaders. Find the number of adult leaders.

Try it.

Huong is organizing paperback and hardback books for her club’s used book sale. The number of paperbacks is 12 less than three times the number of hardbacks. Huong had 162 paperbacks. How many hardback books were there?

Solution

58

Try it.

Jeff is lining up children’s and adult bicycles at the bike shop where he works. The number of children’s bicycles is nine less than three times the number of adult bicycles. There are 42 adult bicycles. How many children’s bicycles are there?

Try it.

Philip pays $1,620 in rent every month. This amount is $120 more than twice what his brother Paul pays for rent. How much does Paul pay for rent?

Solution

$750

Try it.

Marc just bought an SUV for $54,000. This is $7,400 less than twice what his wife paid for her car last year. How much did his wife pay for her car?

Try it.

Laurie has $46,000 invested in stocks and bonds. The amount invested in stocks is $8,000 less than three times the amount invested in bonds. How much does Laurie have invested in bonds?

Solution

$13,500

Try it.

Erica earned a total of $50,450 last year from her two jobs. The amount she earned from her job at the store was $1,250 more than three times the amount she earned from her job at the college. How much did she earn from her job at the college?

Solve Number Problems

In the following exercises, solve each number word problem.

Try it.

The sum of a number and eight is 12. Find the number.

Solution

4

Try it.

The sum of a number and nine is 17. Find the number.

Try it.

The difference of a number and 12 is three. Find the number.

Solution

15

Try it.

The difference of a number and eight is four. Find the number.

Try it.

The sum of three times a number and eight is 23. Find the number.

Solution

5

Try it.

The sum of twice a number and six is 14. Find the number.

Try it.

The difference of twice a number and seven is 17. Find the number.

Solution

12

Try it.

The difference of four times a number and seven is 21. Find the number.

Try it.

Three times the sum of a number and nine is 12. Find the number.

Solution

\(-5\)

Try it.

Six times the sum of a number and eight is 30. Find the number.

Try it.

One number is six more than the other. Their sum is 42. Find the numbers.

Solution

18, 24

Try it.

One number is five more than the other. Their sum is 33. Find the numbers.

Try it.

The sum of two numbers is 20. One number is four less than the other. Find the numbers.

Solution

8, 12

Try it.

The sum of two numbers is 27. One number is seven less than the other. Find the numbers.

Try it.

The sum of two numbers is \(-45.\) One number is nine more than the other. Find the numbers.

Solution

\(-18,-27\)

Try it.

The sum of two numbers is \(-61.\) One number is 35 more than the other. Find the numbers.

Try it.

The sum of two numbers is \(-316.\) One number is 94 less than the other. Find the numbers.

Solution

\(-111,-205\)

Try it.

The sum of two numbers is \(-284.\) One number is 62 less than the other. Find the numbers.

Try it.

One number is 14 less than another. If their sum is increased by seven, the result is 85. Find the numbers.

Solution

32, 46

Try it.

One number is 11 less than another. If their sum is increased by eight, the result is 71. Find the numbers.

Try it.

One number is five more than another. If their sum is increased by nine, the result is 60. Find the numbers.

Solution

23, 28

Try it.

One number is eight more than another. If their sum is increased by 17, the result is 95. Find the numbers.

Try it.

One number is one more than twice another. Their sum is \(-5.\) Find the numbers.

Solution

\(-2,-3\)

Try it.

One number is six more than five times another. Their sum is six. Find the numbers.

Try it.

The sum of two numbers is 14. One number is two less than three times the other. Find the numbers.

Solution

4, 10

Try it.

The sum of two numbers is zero. One number is nine less than twice the other. Find the numbers.

Try it.

The sum of two consecutive integers is 77. Find the integers.

Solution

38, 39

Try it.

The sum of two consecutive integers is 89. Find the integers.

Try it.

The sum of two consecutive integers is \(-23.\) Find the integers.

Solution

\(-11,-12\)

Try it.

The sum of two consecutive integers is \(-37.\) Find the integers.

Try it.

The sum of three consecutive integers is 78. Find the integers.

Solution

25, 26, 27

Try it.

The sum of three consecutive integers is 60. Find the integers.

Try it.

Find three consecutive integers whose sum is \(-36.\)

Solution

\(-11,-12,-13\)

Try it.

Find three consecutive integers whose sum is \(-3.\)

Try it.

Find three consecutive even integers whose sum is 258.

Solution

84, 86, 88

Try it.

Find three consecutive even integers whose sum is 222.

Try it.

Find three consecutive odd integers whose sum is 171.

Solution

55, 57, 59

Try it.

Find three consecutive odd integers whose sum is 291.

Try it.

Find three consecutive even integers whose sum is \(-36.\)

Solution

\(-10,-12,-14\)

Try it.

Find three consecutive even integers whose sum is \(-84.\)

Try it.

Find three consecutive odd integers whose sum is \(-213.\)

Solution

\(-69,-71,-73\)

Try it.

Find three consecutive odd integers whose sum is \(-267.\)

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

1. Translate “six less than twice x” into an algebraic expression.
   If you missed this problem, review .

   Жауап беріңіз

   \(2x-6\)

2. Convert 4.5% to a decimal.
   If you missed this problem, review .

   Жауап беріңіз

   \(0.045\)

3. Convert 0.6 to a percent.
   If you missed this problem, review .

   Жауап беріңіз

   \(60\%\)

4. Normal yearly snowfall at the local ski resort is 12 inches more than twice the amount it received last season. The normal yearly snowfall is 62 inches. What was the snowfall last season at the ski resort?

   Жауап беріңіз

   Step 1. Read the problem.
   Step 2. Identify what you are looking for.	What was the snowfall last season?
   Step 3. Name what we are looking for and choose a variable to represent it.	Let \(s=\) the snowfall last season.
   Step 4. Translate. Restate the problem in one sentence with all the important information.
   Translate into an equation.
   Step 5. Solve the equation.
   Subtract 12 from each side.
   Simplify.
   Divide each side by two.
   Simplify.
   Step 6. Check: First, is our answer reasonable? Yes, having 25 inches of snow seems OK. The problem says the normal snowfall is twelve inches more than twice the number of last season. Twice 25 is 50 and 12 more than that is 62.
   Step 7. Answer the question.	The snowfall last season was 25 inches.

5. Guillermo bought textbooks and notebooks at the bookstore. The number of textbooks was three more than twice the number of notebooks. He bought seven textbooks. How many notebooks did he buy?

   Жауап беріңіз

   He bought two notebooks.

6. Gerry worked Sudoku puzzles and crossword puzzles this week. The number of Sudoku puzzles he completed is eight more than twice the number of crossword puzzles. He completed 22 Sudoku puzzles. How many crossword puzzles did he do?

   Жауап беріңіз

   He did seven crosswords puzzles.

7. The sum of seven times a number and eight is thirty-six. Find the number.

   Жауап беріңіз

   Step 1. Read the problem.
   Step 2. Identify what you are looking for.	the number
   Step 3. Name what you are looking for and choose a variable to represent it.	Let n = the number.
   Step 4. Translate: Restate the problem as one sentence. Translate into an equation.
   Step 5. Solve the equation. Subtract eight from each side and simplify. Divide each side by seven and simplify.
   Step 6. Check. Is the sum of seven times four plus eight equal to 36? \(\ \begin{array}{lll}7\cdot 4+8 & \overset{?}{=} & 36 \\ 28+8 & \overset{?}{=} & 36 \\ 36 & = & 36✓\end{array}\)
   Step 7. Answer the question.	The number is 4.

   Did you notice that we left out some of the steps as we solved this equation? If you’re not yet ready to leave out these steps, write down as many as you need.

8. The sum of four times a number and two is fourteen. Find the number.

   Жауап беріңіз

   \(3\)

9. The sum of three times a number and seven is twenty-five. Find the number.

   Жауап беріңіз

   \(6\)

10. The sum of two numbers is negative fifteen. One number is nine less than the other. Find the numbers.

    Жауап беріңіз

    Step 1. Read the problem.
    Step 2. Identify what you are looking for.	two numbers
    Step 3. Name what you are looking for by choosing a variable to represent the first number. “One number is nine less than the other.”	Let \(n={1}^{\text{st}}\) number. \(n-9={2}^{\text{nd}}\) number
    Step 4. Translate. Write as one sentence. Translate into an equation.	The sum of two numbers is negative fifteen.
    Step 5. Solve the equation. Combine like terms. Add nine to each side and simplify. Simplify.
    Step 6. Check. Is \(-12\) nine less than \(-3?\) \(\ \begin{array}{lll}-3-9 & \overset{?}{=} & -12 \\ -12 & = & -12✓\end{array}\) Is their sum \(-15?\) \(\ \begin{array}{lll}-3+(-12) & \overset{?}{=} & -15 \\ -15 & = & -15✓\end{array}\)
    Step 7. Answer the question.	The numbers are \(-3\) and \(-12.\)

11. The sum of two numbers is negative twenty-three. One number is seven less than the other. Find the numbers.

    Жауап беріңіз

    \(-15,-8\)

12. The sum of two numbers is negative eighteen. One number is forty more than the other. Find the numbers.

    Жауап беріңіз

    \(-29,11\)

13. Find three consecutive integers whose sum is \(-54.\)

    Жауап беріңіз

    Step 1. Read the problem.
    Step 2. Identify what you are looking for.	three consecutive integers
    Step 3. Name each of the three numbers	Let \(n={1}^{\text{st}}\) integer. \(n+1={2}^{\text{nd}}\) consecutive integer \(n+2={3}^{\text{rd}}\) consecutive integer
    Step 4. Translate. Restate as one sentence. Translate into an equation.	The sum of the three integers is \(-54.\)
    Step 5. Solve the equation. Combine like terms. Subtract three from each side. Divide each side by three.
    Step 6. Check. \(\begin{array}{lll} \\ -19+(-18)+(-17) & = & -54 \\ -54 & = & -54✓\end{array}\)
    Step 7. Answer the question.	The three consecutive integers are \(-17,\text{-}\text{18,}\) and \(-19.\)

14. Find three consecutive integers whose sum is \(-96.\)

    Жауап беріңіз

    \(-33,-32,-31\)

15. Find three consecutive integers whose sum is \(-36.\)

    Жауап беріңіз

    \(-13,-12,-11\)

16. Find three consecutive even integers whose sum is \(120\).

    Жауап беріңіз

    Step 1. Read the problem.
    Step 2. Identify what you are looking for.	three consecutive even integers
    Step 3. Name.	\(\begin{array}{l}\text{Let}\ n=\ {1}^{\text{st}}\ \text{even integer.} \\ n+2=\ {2}^{\text{nd}}\ \text{consecutive even integer} \\ n+4=\ {3}^{\text{rd}}\ \text{consecutive even integer}\end{array}\)
    Step 4. Translate. Restate as one sentence. Translate into an equation.	\(\begin{array}{l} \\ \\ \text{The sum of the three even integers is}\ 120. \\ \\ n\ +\ n\ +\ 2\ +\ n+4\ =120\end{array}\)
    Step 5. Solve the equation. Combine like terms. Subtract 6 from each side. Divide each side by 3.	\(\begin{array}{ll} & \ n+n+2+n+4=120 \\ & \ 3n+6=120 \\ & \ 3n=114 \\ & \ n=38\ {1}^{\text{st}}\ \text{integer} \\ \\ & \ n+2\ {2}^{\text{nd}}\ \text{integer} \\ & \ 38+2 \\ & \ 40 \\ \\ & \ n+4\ {3}^{\text{rd}}\ \text{integer} \\ & \ 38+4 \\ & \ 42\end{array}\)
    Step 6. Check. \(\begin{array}{ll} \\ \\ 38+40+42 & \overset{?}{=}120 \\ 120 & =120\ ✓\end{array}\)
    Step 7. Answer the question.	The three consecutive integers are 38, 40, and 42.

17. Find three consecutive even integers whose sum is 102.

    Жауап беріңіз

    32, 34, 36

18. Find three consecutive even integers whose sum is \(-24.\)

    Жауап беріңіз

    \(-10,-8,-6\)

19. A married couple together earns $110,000 a year. The wife earns $16,000 less than twice what her husband earns. What does the husband earn?

    Жауап беріңіз

    Step 1. Read the problem.
    Step 2. Identify what you are looking for.	How much does the husband earn?
    Step 3. Name. Choose a variable to represent the amount the husband earns. The wife earns $16,000 less than twice that.	Let h = the amount the husband earns.
    Step 4. Translate. Restate the problem in one sentence with all the important information. Translate into an equation.	2h − 16,000 = the amount the wife Together the husband and wife earn $110,000.
    Step 5. Solve the equation. Combine like terms. Add 16,000 to both sides and simplify. Divide each side by three.	\(\begin{array}{lll}h+2h-16,000 & = & 110,000 \\ h+2h-16,000 & = & 110,000 \\ 3h-16,000 & = & 110,000 \\ 3h & = & 126,000 \\ h & = & 42,000\end{array}\)
    	$42,000 amount husband earns
    	\(\begin{array}{l}2h-16,000\ \text{amount wife earns} \\ 2(42,000)-16,000 \\ 84,000-16,000 \\ 68,000\end{array}\)
    Step 6. Check: If the wife earns $68,000 and the husband earns $42,000, is that $110,000? Yes!
    Step 7. Answer the question.	The husband earns $42,000 a year.

20. According to the National Automobile Dealers Association, the average cost of a car in 2014 was $28,400. This was $1,600 less than six times the cost in 1975. What was the average cost of a car in 1975?

    Жауап беріңіз

    The average cost was $5,000.

21. US Census data shows that the median price of new home in the U.S. in November 2014 was $280,900. This was $10,700 more than 14 times the price in November 1964. What was the median price of a new home in November 1964?

    Жауап беріңіз

    The median price was $19,300.

22. Translate and solve:

    ⓐ What number is 45% of 84?ⓑ 8.5% of what amount is $4.76? ⓒ 168 is what percent of 112?

    Жауап беріңіз

    ⓐ

    Translate into algebra. Let n = the number.
    Multiply.
    			37.8 is 45% of 84.

    
    
    ⓑ
    

    Translate. Let n = the amount.
    Multiply.
    Divide both sides by 0.085 and simplify.
    			8.5% of $56 is $4.76

    
    
    ⓒ
    

    We are asked to find percent, so we must have our result in percent form.
    Translate into algebra. Let p = the percent.
    Multiply.
    Divide both sides by 112 and simplify.
    Convert to percent.
    			168 is 150% of 112.

23. Translate and solve: ⓐ What number is 45% of 80? ⓑ 7.5% of what amount is $1.95? ⓒ 110 is what percent of 88?

    Жауап беріңіз

    ⓐ 36 ⓑ $26 ⓒ \(125\%\)

24. Translate and solve: ⓐ What number is 55% of 60? ⓑ 8.5% of what amount is $3.06? ⓒ 126 is what percent of 72?

    Жауап беріңіз

    ⓐ 33 ⓑ $36 ⓒ \(175\%\)

25. The label on Audrey’s yogurt said that one serving provided 12 grams of protein, which is 24% of the recommended daily amount. What is the total recommended daily amount of protein?

    Жауап беріңіз

    What are you asked to find?	What total amount of protein is recommended?
    Choose a variable to represent it.	Let \(a=\) total amount of protein.
    Write a sentence that gives the information to find it.
    Translate into an equation.
    Solve.
    Check: Does this make sense? Yes, 24% is about \(\frac{1}{4}\) of the total and 12 is about \(\frac{1}{4}\) of 50.
    Write a complete sentence to answer the question.	The amount of protein that is recommended is 50 g.

26. One serving of wheat square cereal has 7 grams of fiber, which is 28% of the recommended daily amount. What is the total recommended daily amount of fiber?

    Жауап беріңіз

    25 grams

27. One serving of rice cereal has 190 mg of sodium, which is 8% of the recommended daily amount. What is the total recommended daily amount of sodium?

    Жауап беріңіз

    2,375 mg

28. Veronica is planning to make muffins from a mix. The package says each muffin will be 240 calories and 60 calories will be from fat. What percent of the total calories is from fat?

    Жауап беріңіз

    What are you asked to find?	What percent of the total calories is fat?
    Choose a variable to represent it.	Let \(p=\) percent of fat.
    Write a sentence that gives the information to find it.
    Translate the sentence into an equation.
    Multiply.
    Divide both sides by 240.
    Put in percent form.
    Check: does this make sense? Yes, \(25\%\) is one-fourth; 60 is one-fourth of 240. So, \(25\%\) makes sense.
    Write a complete sentence to answer the question.	Of the total calories in each muffin, \(25\%\) is fat.

29. Mitzi received some gourmet brownies as a gift. The wrapper said each brownie was 480 calories, and had 240 calories of fat. What percent of the total calories in each brownie comes from fat? Round the answer to the nearest whole percent.

    Жауап беріңіз

    50%

30. The mix Ricardo plans to use to make brownies says that each brownie will be 190 calories, and 76 calories are from fat. What percent of the total calories are from fat? Round the answer to the nearest whole percent.

    Жауап беріңіз

    40%

31. Recently, the California governor proposed raising community college fees from $36 a unit to $46 a unit. Find the percent change. (Round to the nearest tenth of a percent.)

    Жауап беріңіз

    Find the amount of change.	\(46-36=10\)
    Find the percent.	Change is what percent of the original amount?
    Let \(p=\) the percent.
    Translate to an equation.
    Simplify.
    Divide both sides by 36.
    Change to percent form; round to the nearest tenth
    Write a complete sentence to answer the question.	The new fees are approximately a \(27.8\%\) increase over the old fees.
    Remember to round the division to the nearest thousandth in order to round the percent to the nearest tenth.

32. Find the percent change. (Round to the nearest tenth of a percent.) In 2011, the IRS increased the deductible mileage cost to 55.5 cents from 51 cents.

    Жауап беріңіз

    \(8.8\%\)

33. Find the percent change. (Round to the nearest tenth of a percent.) In 1995, the standard bus fare in Chicago was $1.50. In 2008, the standard bus fare was 2.25.

    Жауап беріңіз

    50%

34. Liam’s art gallery bought a painting at an original cost of $750. Liam marked the price up 40%. Find ⓐ the amount of mark-up and ⓑ the list price of the painting.

    Жауап беріңіз

    ⓐ
    
    

    Identify what you are asked to find, and choose a variable to represent it.	What is the amount of mark-up? Let \(m=\) the amount of mark-up.
    Write a sentence that gives the information to find it.
    Translate into an equation.
    Solve the equation.
    Write a complete sentence.	The mark-up on the painting was $300.

    
    
    ⓑ
    

    Identify what you are asked to find, and choose a variable to represent it.	What is the list price? Let \(p=\) the list price.
    Write a sentence that gives the information to find it.
    Translate into an equation.
    Solve the equation.
    Check.	Is the list price more than the original cost? Is $1,050 more than $750? Yes.
    Write a complete sentence.	The list price of the painting was $1,050.

35. Find ⓐ the amount of mark-up and ⓑ the list price: Jim’s music store bought a guitar at original cost $1,200. Jim marked the price up 50%.

    Жауап беріңіз

    ⓐ $600 ⓑ $1,800

36. Find ⓐ the amount of mark-up and ⓑ the list price: The Auto Resale Store bought Pablo’s Toyota for $8,500. They marked the price up 35%.

    Жауап беріңіз

    ⓐ $2,975 ⓑ $11,475

37. Areli invested a principal of $950 in her bank account that earned simple interest at an interest rate of 3%. How much interest did she earn in five years?

    Жауап беріңіз

    \(\begin{array}{lll}I & = & ? \\ P & = & \text{\$}950 \\ r & = & 3\% \\ t & = & 5\ \text{years}\end{array}\)

    Identify what you are asked to find, and choose a variable to represent it.	What is the simple interest? \(\text{Let}\ I=\text{interest.}\)
    Write the formula.	\(I=Prt\)
    Substitute in the given information.	\(I=(950)(0.03)(5)\)
    Simplify.	\(I=142.5\)
    Check.
    Is $142.50 a reasonable amount of interest on $950?
    Yes.
    Write a complete sentence.	The interest is $142.50.

38. Nathaly deposited $12,500 in her bank account where it will earn 4% simple interest. How much interest will Nathaly earn in five years?

    Жауап беріңіз

    He will earn $2,500.

39. Susana invested a principal of $36,000 in her bank account that earned simple interest at an interest rate of \(6.5\%.\) How much interest did she earn in three years?

    Жауап беріңіз

    She earned $7,020.

40. Hang borrowed $7,500 from her parents to pay her tuition. In five years, she paid them $1,500 interest in addition to the $7,500 she borrowed. What was the rate of simple interest?

    Жауап беріңіз

    \(\begin{array}{lll}I & = & \text{\$}1500 \\ P & = & \text{\$}7500 \\ r & = & ? \\ t & = & 5\ \text{years}\end{array}\)

    Identify what you are asked to find, and choose a variable to represent it.	What is the rate of simple interest? \(\text{Let}\ r=\text{rate of interest.}\)
    Write the formula. Substitute in the given information. Multiply. Divide. Change to percent form.	\(\begin{array}{lll}I & = & Prt \\ 1,500 & = & (7,500)r(5) \\ 1,500 & = & 37,500r \\ 0.04 & = & r \\ 4\% & = & r\end{array}\)
    Check. \(\begin{array}{lll}I & = & Prt \\ 1,500 & \overset{?}{=} & (7,500)(0.04)(5) \\ 1,500 & = & 1,500✓\end{array}\)
    Write a complete sentence.	The rate of interest was 4%.

Symbols used here

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\neq

not equal
The two sides are different.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

|x|

absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.

i

imaginary unit
i² = −1.

\log_b x,\ \ln x

logarithm, natural log
The exponent b must be raised to for x; ln uses base e.

\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}

number sets
Naturals, integers, rationals, reals, complex numbers.

How to: Use a Problem Solving Strategy

1. Use a problem solving strategy for word problems
2. Solve number word problems
3. Solve percent applications
4. Solve simple interest applications
5. Find the amount of change.
6. Find what percent the amount of change is of the original amount.
7. Find the amount of change
8. Find what percent the amount of change is of the original amount.

Questions people ask

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

Өзіңіздіңіңізді сынап көріңіз

⌨ Type ✎ Draw

Математикалық тапсырма

& Шегіну

x²√÷×π( )=∫d/dxlim[ ]sinlogeˣ

Parts of this page are adapted from OpenStax Elementary Algebra 2e (CC BY-NC-SA 4.0), OpenStax Intermediate Algebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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