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Solve Quadratic Equations Using the Quadratic Formula

Solve quadratic equations using the Quadratic Formula

Solve Quadratic Equations Using the Quadratic Formula

When we solved quadratic equations in the last section by completing the square, we took the same steps every time. By the end of the exercise set, you may have been wondering ‘isn’t there an easier way to do this?’ The answer is ‘yes’. Mathematicians look for patterns when they do things over and over in order to make their work easier. In this section we will derive and use a formula to find the solution of a quadratic equation.

We have already seen how to solve a formula for a specific variable ‘in general’, so that we would do the algebraic steps only once, and then use the new formula to find the value of the specific variable. Now we will go through the steps of completing the square using the general form of a quadratic equation to solve a quadratic equation for x.

We start with the standard form of a quadratic equation and solve it for x by completing the square.


Isolate the variable terms on one side.
Make the coefficient of \({x}^{2}\) equal to 1, by dividing by a.
Simplify.
To complete the square, find \({(\frac{1}{2}\cdot \frac{b}{a})}^{2}\) and add it to both sides of the equation.
\({(\frac{1}{2}\frac{b}{a})}^{2}=\frac{{b}^{2}}{4{a}^{2}}\)
The left side is a perfect square, factor it.
Find the common denominator of the right side and write equivalent fractions with the common denominator.
Simplify.
Combine to one fraction.
Use the square root property.
Simplify the radical.
Add \(-\frac{b}{2a}\) to both sides of the equation.
Combine the terms on the right side.
	This equation is the Quadratic Formula.

To use the Quadratic Formula, we substitute the values of a, b, and c from the standard form into the expression on the right side of the formula. Then we simplify the expression. The result is the pair of solutions to the quadratic equation.

Notice the formula is an equation. Make sure you use both sides of the equation.

How to Solve a Quadratic Equation Using the Quadratic Formula

Try it.

Solve by using the Quadratic Formula: \(2{x}^{2}+9x-5=0.\)

Solution

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

Use the Discriminant to Predict the Number and Type of Solutions of a Quadratic Equation

When we solved the quadratic equations in the previous examples, sometimes we got two real solutions, one real solution, and sometimes two complex solutions. Is there a way to predict the number and type of solutions to a quadratic equation without actually solving the equation?

Yes, the expression under the radical of the Quadratic Formula makes it easy for us to determine the number and type of solutions. This expression is called the discriminant.

Let’s look at the discriminant of the equations in some of the examples and the number and type of solutions to those quadratic equations.

Quadratic Equation (in standard form)	Discriminant \({b}^{2}-4ac\)	Value of the Discriminant	Number and Type of solutions
\(2{x}^{2}+9x-5=0\)	\(\begin{array}{l}{9}^{2}-4\cdot 2(-5) \\ 121\end{array}\)	+	2 real
\(4{x}^{2}-20x+25=0\)	\(\begin{array}{l}{(-20)}^{2}-4\cdot 4\cdot 25 \\ 0\end{array}\)	0	1 real
\(3{p}^{2}+2p+9=0\)	\(\begin{array}{l}{2}^{2}-4\cdot 3\cdot 9 \\ \\ -104\end{array}\)	−	2 complex

Example

Try it.

Determine the number of solutions to each quadratic equation.

ⓐ \(3{x}^{2}+7x-9=0\) ⓑ \(5{n}^{2}+n+4=0\) ⓒ \(9{y}^{2}-6y+1=0.\)

Solution

To determine the number of solutions of each quadratic equation, we will look at its discriminant.

ⓐ

	\(3{x}^{2}+7x-9=0\)
The equation is in standard form, identify a, b, and c.	\(a=3,\ b=7,\ c=-9\)
Write the discriminant.	\({b}^{2}-4ac\)
Substitute in the values of a, b, and c.	\({(7)}^{2}-4\cdot 3\cdot (-9)\)
Simplify.	\(49+108\)
	\(157\)

Since the discriminant is positive, there are 2 real solutions to the equation.

ⓑ

	\(5{n}^{2}+n+4=0\)
The equation is in standard form, identify a, b, and c.	\(a=5,\ b=1,\ c=4\)
Write the discriminant.	\({b}^{2}-4ac\)
Substitute in the values of a, b, and c.	\({(1)}^{2}-4\cdot 5\cdot 4\)
Simplify.	\(1-80\)
	\(-79\)

Since the discriminant is negative, there are 2 complex solutions to the equation.

ⓒ

	\(9{y}^{2}-6y+1=0\)
The equation is in standard form, identify a, b, and c.	\(a=9,b=-6,c=1\)
Write the discriminant.	\({b}^{2}-4ac\)
Substitute in the values of a, b, and c.	\({(-6)}^{2}-4\cdot 9\cdot 1\)
Simplify.	\(36-36\)
	\(0\)

Since the discriminant is 0, there is 1 real solution to the equation.

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

Identify the Most Appropriate Method to Use to Solve a Quadratic Equation

We summarize the four methods that we have used to solve quadratic equations below.

Given that we have four methods to use to solve a quadratic equation, how do you decide which one to use? Factoring is often the quickest method and so we try it first. If the equation is \(a{x}^{2}=k\) or \(a{(x-h)}^{2}=k\) we use the Square Root Property. For any other equation, it is probably best to use the Quadratic Formula. Remember, you can solve any quadratic equation by using the Quadratic Formula, but that is not always the easiest method.

What about the method of Completing the Square? Most people find that method cumbersome and prefer not to use it. We needed to include it in the list of methods because we completed the square in general to derive the Quadratic Formula. You will also use the process of Completing the Square in other areas of algebra.

The next example uses this strategy to decide how to solve each quadratic equation.

Example

Try it.

Identify the most appropriate method to use to solve each quadratic equation.

ⓐ \(5{z}^{2}=17\) ⓑ \(4{x}^{2}-12x+9=0\) ⓒ \(8{u}^{2}+6u=11.\)

Solution

ⓐ
\(\begin{array}{l} \\ \\ \\ 5{z}^{2}=17\end{array}\)

Since the equation is in the \(a{x}^{2}=k,\) the most appropriate method is to use the Square Root Property.

ⓑ
\(\begin{array}{l} \\ \\ \\ 4{x}^{2}-12x+9=0\end{array}\)

We recognize that the left side of the equation is a perfect square trinomial, and so factoring will be the most appropriate method.

ⓒ

	\(8{u}^{2}+6u=11\)
Put the equation in standard form.	\(8{u}^{2}+6u-11=0\)

While our first thought may be to try factoring, thinking about all the possibilities for trial and error method leads us to choose the Quadratic Formula as the most appropriate method.

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

Key Concepts

Quadratic Formula
- The solutions to a quadratic equation of the form ax² + bx + c = 0, \(a\ne 0\) are given by the formula:
  \[x=\frac{\text{-}b\pm \sqrt{{b}^{2}-4ac}}{2a}\]
How to solve a quadratic equation using the Quadratic Formula.
1. Write the quadratic equation in standard form, ax² + bx + c = 0. Identify the values of a, b, c.
2. Write the Quadratic Formula. Then substitute in the values of a, b, c.
3. Simplify.
4. Check the solutions.
Using the Discriminant, b² − 4ac, to Determine the Number and Type of Solutions of a Quadratic Equation
- For a quadratic equation of the form ax² + bx + c = 0, \(a\ne 0,\)
  - If b² − 4ac > 0, the equation has 2 real solutions.
  - if b² − 4ac = 0, the equation has 1 real solution.
  - if b² − 4ac < 0, the equation has 2 complex solutions.
Methods to Solve Quadratic Equations:
- Factoring
- Square Root Property
- Completing the Square
- Quadratic Formula
How to identify the most appropriate method to solve a quadratic equation.
1. Try Factoring first. If the quadratic factors easily, this method is very quick.
2. Try the Square Root Property next. If the equation fits the form ax² = k or a(x − h)² = k, it can easily be solved by using the Square Root Property.
3. Use the Quadratic Formula. Any other quadratic equation is best solved by using the Quadratic Formula.

Solve Quadratic Equations Using the Quadratic Formula

Solve Quadratic Equations Using the Quadratic Formula

In the following exercises, solve by using the Quadratic Formula.

Try it.

\(4{m}^{2}+m-3=0\)

Solution

\(m=-1,m=\frac{3}{4}\)

Try it.

\(4{n}^{2}-9n+5=0\)

Try it.

\(2{p}^{2}-7p+3=0\)

Solution

\(p=\frac{1}{2},p=3\)

Try it.

\(3{q}^{2}+8q-3=0\)

Try it.

\({p}^{2}+7p+12=0\)

Solution

\(p=-4,p=-3\)

Try it.

\({q}^{2}+3q-18=0\)

Try it.

\({r}^{2}-8r=33\)

Solution

\(r=-3,r=11\)

Try it.

\({t}^{2}+13t=-40\)

Try it.

\(3{u}^{2}+7u-2=0\)

Solution

\(u=\frac{-7\pm \sqrt{73}}{6}\)

Try it.

\(2{p}^{2}+8p+5=0\)

Try it.

\(2{a}^{2}-6a+3=0\)

Solution

\(a=\frac{3\pm \sqrt{3}}{2}\)

Try it.

\(5{b}^{2}+2b-4=0\)

Try it.

\({x}^{2}+8x-4=0\)

Solution

\(x=-4\pm 2\sqrt{5}\)

Try it.

\({y}^{2}+4y-4=0\)

Try it.

\(3{y}^{2}+5y-2=0\)

Solution

\(y=-2,y=\frac{1}{3}\)

Try it.

\(6{x}^{2}+2x-20=0\)

Try it.

\(2{x}^{2}+3x+3=0\)

Solution

\(x=-\frac{3}{4}\pm \frac{\sqrt{15}}{4}i\)

Try it.

\(2{x}^{2}-x+1=0\)

Try it.

\(8{x}^{2}-6x+2=0\)

Solution

\(x=\frac{3}{8}\pm \frac{\sqrt{7}}{8}i\)

Try it.

\(8{x}^{2}-4x+1=0\)

Try it.

\((v+1)(v-5)-4=0\)

Solution

\(v=2\pm \sqrt{13}\)

Try it.

\((x+1)(x-3)=2\)

Try it.

\((y+4)(y-7)=18\)

Solution

\(y=\frac{3\pm \sqrt{193}}{2}\)

Try it.

\((x+2)(x+6)=21\)

Try it.

\(\frac{1}{3}{m}^{2}+\frac{1}{12}m=\frac{1}{4}\)

Solution

\(m=-1,m=\frac{3}{4}\)

Try it.

\(\frac{1}{3}{n}^{2}+n=-\frac{1}{2}\)

Try it.

\(\frac{3}{4}{b}^{2}+\frac{1}{2}b=\frac{3}{8}\)

Solution

\(b=\frac{-2\pm \sqrt{22}}{6}\)

Try it.

\(\frac{1}{9}{c}^{2}+\frac{2}{3}c=3\)

Try it.

\(16{c}^{2}+24c+9=0\)

Solution

\(c=-\frac{3}{4}\)

Try it.

\(25{d}^{2}-60d+36=0\)

Try it.

\(25{q}^{2}+30q+9=0\)

Solution

\(q=-\frac{3}{5}\)

Try it.

\(16{y}^{2}+8y+1=0\)

Use the Discriminant to Predict the Number of Real Solutions of a Quadratic Equation

In the following exercises, determine the number of real solutions for each quadratic equation.

Try it.

ⓐ \(4{x}^{2}-5x+16=0\) ⓑ \(36{y}^{2}+36y+9=0\) ⓒ \(6{m}^{2}+3m-5=0\)

Solution

ⓐ \(\text{no real solutions}\) ⓑ \(1\)
ⓒ \(2\)

Try it.

ⓐ \(9{v}^{2}-15v+25=0\) ⓑ \(100{w}^{2}+60w+9=0\) ⓒ \(5{c}^{2}+7c-10=0\)

Try it.

ⓐ \({r}^{2}+12r+36=0\) ⓑ \(8{t}^{2}-11t+5=0\) ⓒ \(3{v}^{2}-5v-1=0\)

Solution

ⓐ \(1\) ⓑ \(\text{no real solutions}\)
ⓒ \(2\)

Try it.

ⓐ \(25{p}^{2}+10p+1=0\) ⓑ \(7{q}^{2}-3q-6=0\) ⓒ \(7{y}^{2}+2y+8=0\)

Identify the Most Appropriate Method to Use to Solve a Quadratic Equation

In the following exercises, identify the most appropriate method (Factoring, Square Root, or Quadratic Formula) to use to solve each quadratic equation. Do not solve.

Try it.

ⓐ \({x}^{2}-5x-24=0\)
ⓑ \({(y+5)}^{2}=12\)
ⓒ \(14{m}^{2}+3m=11\)

Solution

ⓐ \(\text{factor}\)
ⓑ \(\text{square root}\)
ⓒ \(\text{Quadratic Formula}\)

Try it.

ⓐ \({(8v+3)}^{2}=81\)
ⓑ \({w}^{2}-9w-22=0\)
ⓒ \(4{n}^{2}-10n=6\)

Try it.

ⓐ \(6{a}^{2}+14a=20\)
ⓑ \({(x-\frac{1}{4})}^{2}=\frac{5}{16}\)
ⓒ \({y}^{2}-2y=8\)

Solution

ⓐ \(\text{Quadratic Formula}\)
ⓑ \(\text{square root}\)
ⓒ \(\text{factor}\)

Try it.

ⓐ \(8{b}^{2}+15b=4\)
ⓑ \(\frac{5}{9}{v}^{2}-\frac{2}{3}v=1\)
ⓒ \({(w+\frac{4}{3})}^{2}=\frac{2}{9}\)

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

Use the Discriminant to Predict the Number of Solutions of a Quadratic Equation

When we solved the quadratic equations in the previous examples, sometimes we got two solutions, sometimes one solution, sometimes no real solutions. Is there a way to predict the number of solutions to a quadratic equation without actually solving the equation?

Yes, the quantity inside the radical of the Quadratic Formula makes it easy for us to determine the number of solutions. This quantity is called the discriminant.

Let’s look at the discriminant of the equations in , , and , and the number of solutions to those quadratic equations.

Quadratic Equation (in standard form)	Discriminant \({b}^{2}-4ac\)	Sign of the Discriminant	Number of real solutions
\(2{x}^{2}+9x-5=0\)	\({9}^{2}-4\cdot 2(-5)=121\)	+	2
\(4{x}^{2}-20x+25=0\)	\({(-20)}^{2}-4\cdot 4\cdot 25=0\)	0	1
\(3{p}^{2}+2p+9=0\)	\({2}^{2}-4\cdot 3\cdot 9=-104\)	−	0

When the discriminant is positive \((x=\frac{\text{-}b\pm \sqrt{+}}{2a})\) the quadratic equation has two solutions.

When the discriminant is zero \((x=\frac{\text{-}b\pm \sqrt{0}}{2a})\) the quadratic equation has one solution.

When the discriminant is negative \((x=\frac{\text{-}b\pm \sqrt{-}}{2a})\) the quadratic equation has no real solutions.

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

Identify the Most Appropriate Method to Use to Solve a Quadratic Equation

We have used four methods to solve quadratic equations:

Factoring
Square Root Property
Completing the Square
Quadratic Formula

You can solve any quadratic equation by using the Quadratic Formula, but that is not always the easiest method to use.

What about the method of completing the square? Most people find that method cumbersome and prefer not to use it. We needed to include it in this chapter because we completed the square in general to derive the Quadratic Formula. You will also use the process of completing the square in other areas of algebra.

Example

Try it.

Identify the most appropriate method to use to solve each quadratic equation:

ⓐ \(5{z}^{2}=17\) ⓑ \(4{x}^{2}-12x+9=0\) ⓒ \(8{u}^{2}+6u=11\)

Solution

ⓐ \(5{z}^{2}=17\)

Since the equation is in the \(a{x}^{2}=k\), the most appropriate method is to use the Square Root Property.

ⓑ \(4{x}^{2}-12x+9=0\)

We recognize that the left side of the equation is a perfect square trinomial, and so Factoring will be the most appropriate method.

Put the equation in standard form. \(8{u}^{2}+6u-11=0\)

While our first thought may be to try Factoring, thinking about all the possibilities for trial and error leads us to choose the Quadratic Formula as the most appropriate method

Key Concepts

Quadratic Formula The solutions to a quadratic equation of the form \(a{x}^{2}+bx+c=0,\) \(a\ne 0\) are given by the formula:
\[x=\frac{\text{-}b\pm \sqrt{{b}^{2}-4ac}}{2a}\]
Solve a Quadratic Equation Using the Quadratic Formula
To solve a quadratic equation using the Quadratic Formula.
1. Write the quadratic formula in standard form. Identify the \(a,b,c\) values.
2. Write the quadratic formula. Then substitute in the values of \(a,b,c.\)
3. Simplify.
4. Check the solutions.
Using the Discriminant, \({b}^{2}-4ac\), to Determine the Number of Solutions of a Quadratic Equation
For a quadratic equation of the form \(a{x}^{2}+bx+c=0,\) \(a\ne 0,\)
- if \({b}^{2}-4ac>0\), the equation has 2 solutions.
- if \({b}^{2}-4ac=0\), the equation has 1 solution.
- if \({b}^{2}-4ac<0\), the equation has no real solutions.
To identify the most appropriate method to solve a quadratic equation:
1. Try Factoring first. If the quadratic factors easily this method is very quick.
2. Try the Square Root Property next. If the equation fits the form \(a{x}^{2}=k\) or \(a{(x-h)}^{2}=k\), it can easily be solved by using the Square Root Property.
3. Use the Quadratic Formula. Any other quadratic equation is best solved by using the Quadratic Formula.

Solve Quadratic Equations Using the Quadratic Formula

Solve Quadratic Equations Using the Quadratic Formula

In the following exercises, solve by using the Quadratic Formula.

Try it.

\(4{m}^{2}+m-3=0\)

Solution

\(m=-1,m=\frac{3}{4}\)

Try it.

\(4{n}^{2}-9n+5=0\)

Try it.

\(2{p}^{2}-7p+3=0\)

Solution

\(p=\frac{1}{2},p=3\)

Try it.

\(3{q}^{2}+8q-3=0\)

Try it.

\({p}^{2}+7p+12=0\)

Solution

\(p=-4,p=-3\)

Try it.

\({q}^{2}+3q-18=0\)

Try it.

\({r}^{2}-8r-33=0\)

Solution

\(r=-3,r=11\)

Try it.

\({t}^{2}+13t+40=0\)

Try it.

\(3{u}^{2}+7u-2=0\)

Solution

\(u=\frac{-7\pm \sqrt{73}}{6}\)

Try it.

\(6{z}^{2}-9z+1=0\)

Try it.

\(2{a}^{2}-6a+3=0\)

Solution

\(a=\frac{3\pm \sqrt{3}}{2}\)

Try it.

\(5{b}^{2}+2b-4=0\)

Try it.

\(2{x}^{2}+3x+9=0\)

Solution

no real solution

Try it.

\(6{y}^{2}-5y+2=0\)

Try it.

\(v(v+5)-10=0\)

Solution

\(v=\frac{-5\pm \sqrt{65}}{2}\)

Try it.

\(3w(w-2)-8=0\)

Try it.

\(\frac{1}{3}{m}^{2}+\frac{1}{12}m=\frac{1}{4}\)

Solution

\(m=-1,m=\frac{3}{4}\)

Try it.

\(\frac{1}{3}{n}^{2}+n=-\frac{1}{2}\)

Try it.

\(16{c}^{2}+24c+9=0\)

Solution

\(c=-\frac{3}{4}\)

Try it.

\(25{d}^{2}-60d+36=0\)

Try it.

\(5{m}^{2}+2m-7=0\)

Solution

\(m=-\frac{7}{5},m=1\)

Try it.

\(8{n}^{2}-3n+3=0\)

Try it.

\({p}^{2}-6p-27=0\)

Solution

\(p=-3,p=9\)

Try it.

\(25{q}^{2}+30q+9=0\)

Try it.

\(4{r}^{2}+3r-5=0\)

Solution

\(r=\frac{-3\pm \sqrt{89}}{8}\)

Try it.

\(3t(t-2)=2\)

Try it.

\(2{a}^{2}+12a+5=0\)

Solution

\(a=\frac{-6\pm \sqrt{26}}{2}\)

Try it.

\(4{d}^{2}-7d+2=0\)

Try it.

\(\frac{3}{4}{b}^{2}+\frac{1}{2}b=\frac{3}{8}\)

Solution

\(b=\frac{-2\pm \sqrt{22}}{6}\)

Try it.

\(\frac{1}{9}{c}^{2}+\frac{2}{3}c=3\)

Try it.

\(2{x}^{2}+12x-3=0\)

Solution

\(x=\frac{-6\pm \sqrt{42}}{4}\)

Try it.

\(16{y}^{2}+8y+1=0\)

Use the Discriminant to Predict the Number of Solutions of a Quadratic Equation

In the following exercises, determine the number of solutions to each quadratic equation.

Try it.

ⓐ \(4{x}^{2}-5x+16=0\)
ⓑ \(36{y}^{2}+36y+9=0\)
ⓓ \(18{n}^{2}-7n+3=0\)

Solution

Try it.

ⓐ \(9{v}^{2}-15v+25=0\)
ⓑ \(100{w}^{2}+60w+9=0\)
ⓓ \(15{d}^{2}-4d+8=0\)

Try it.

ⓐ \({r}^{2}+12r+36=0\)
ⓑ \(8{t}^{2}-11t+5=0\)
ⓓ \(3{v}^{2}-5v-1=0\)

Solution

Try it.

ⓐ \(25{p}^{2}+10p+1=0\)
ⓑ \(7{q}^{2}-3q-6=0\)
ⓓ \(25{z}^{2}-60z+36=0\)

Identify the Most Appropriate Method to Use to Solve a Quadratic Equation

In the following exercises, identify the most appropriate method (Factoring, Square Root, or Quadratic Formula) to use to solve each quadratic equation. Do not solve.

Try it.

Solution

Try it.

Solution

Try it.

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.

Evaluate \({b}^{2}-4ab\) when \(a=3\) and \(b=-2.\)
If you missed this problem, review .

Révèle la réponse

\(28\)
Simplify: \(\sqrt{108}.\)
If you missed this problem, review .

Révèle la réponse

\(6\sqrt{3}\)
Simplify: \(\sqrt{50}.\)
If you missed this problem, review .

Révèle la réponse

\(5\sqrt{2}\)
Solve by using the Quadratic Formula: \(2{x}^{2}+9x-5=0.\)
Solve by using the Quadratic Formula: \(3{y}^{2}-5y+2=0\).

Révèle la réponse

\(y=1,y=\frac{2}{3}\)
Solve by using the Quadratic Formula: \(4{z}^{2}+2z-6=0\).

Révèle la réponse

\(z=1,z=-\frac{3}{2}\)

Solve by using the Quadratic Formula: \({x}^{2}-6x=-5.\)

Révèle la réponse


Write the equation in standard form by adding 5 to each side.
This equation is now in standard form.
Identify the values of \(a,\ b,\ c.\)
Write the Quadratic Formula.
Then substitute in the values of \(a,\ b,\ c.\)
Simplify.
Rewrite to show two solutions.
Simplify.

Check:

Solve by using the Quadratic Formula: \({a}^{2}-2a=15\).

Révèle la réponse

\(a=-3,a=5\)
Solve by using the Quadratic Formula: \({b}^{2}+24=-10b\).

Révèle la réponse

\(b=-6,b=-4\)

Solve by using the Quadratic Formula: \(2{x}^{2}+10x+11=0.\)

Révèle la réponse


This equation is in standard form.
Identify the values of a, b, and c.
Write the Quadratic Formula.
Then substitute in the values of a, b, and c.
Simplify.

Simplify the radical.
Factor out the common factor in the numerator.
Remove the common factors.
Rewrite to show two solutions.
Check: We leave the check for you!

Solve by using the Quadratic Formula: \(3{m}^{2}+12m+7=0\).

Révèle la réponse

\(m=\frac{-6+\sqrt{15}}{3},m=\frac{-6-\sqrt{15}}{3}\)
Solve by using the Quadratic Formula: \(5{n}^{2}+4n-4=0\).

Révèle la réponse

\(n=\frac{-2+2\sqrt{6}}{5},n=\frac{-2-2\sqrt{6}}{5}\)

Solve by using the Quadratic Formula: \(3{p}^{2}+2p+9=0.\)

Révèle la réponse


This equation is in standard form
Identify the values of \(a,b,c.\)
Write the Quadratic Formula.
Then substitute in the values of \(a,b,c\).
Simplify.

Simplify the radical using complex numbers.
Simplify the radical.
Factor the common factor in the numerator.
Remove the common factors.
Rewrite in standard \(a+bi\) form.
Write as two solutions.

Solve by using the Quadratic Formula: \(4{a}^{2}-2a+8=0\).

Révèle la réponse

\(a=\frac{1}{4}+\frac{\sqrt{31}}{4}i,\ \ a=\frac{1}{4}-\frac{\sqrt{31}}{4}i\)
Solve by using the Quadratic Formula: \(5{b}^{2}+2b+4=0\).

Révèle la réponse

\(b=-\frac{1}{5}+\frac{\sqrt{19}}{5}i,\ \ b=-\frac{1}{5}-\frac{\sqrt{19}}{5}i\)

Solve by using the Quadratic Formula: \(x(x+6)+4=0.\)

Révèle la réponse

Our first step is to get the equation in standard form.


Distribute to get the equation in standard form.
This equation is now in standard form
Identify the values of \(a,b,c.\)
Write the Quadratic Formula.
Then substitute in the values of \(a,b,c\).
Simplify.

Simplify the radical.
Factor the common factor in the numerator.
Remove the common factors.
Write as two solutions.
Check: We leave the check for you!

Solve by using the Quadratic Formula: \(x(x+2)-5=0.\)

Révèle la réponse

\(x=-1+\sqrt{6},x=-1-\sqrt{6}\)
Solve by using the Quadratic Formula: \(3y(y-2)-3=0.\)

Révèle la réponse

\(y=1+\sqrt{2},y=1-\sqrt{2}\)

Solve by using the Quadratic Formula: \(\frac{1}{2}{u}^{2}+\frac{2}{3}u=\frac{1}{3}.\)

Révèle la réponse

Our first step is to clear the fractions.


Multiply both sides by the LCD, 6, to clear the fractions.
Multiply.
Subtract 2 to get the equation in standard form.
Identify the values of a, b, and c.
Write the Quadratic Formula.
Then substitute in the values of a, b, and c.
Simplify.

Simplify the radical.
Factor the common factor in the numerator.
Remove the common factors.
Rewrite to show two solutions.
Check: We leave the check for you!

Solve by using the Quadratic Formula: \(\frac{1}{4}{c}^{2}-\frac{1}{3}c=\frac{1}{12}\).

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\(c=\frac{2+\sqrt{7}}{3},\ \ c=\frac{2-\sqrt{7}}{3}\)
Solve by using the Quadratic Formula: \(\frac{1}{9}{d}^{2}-\frac{1}{2}d=-\frac{1}{3}\).

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\(d=\frac{9+\sqrt{33}}{4},\ \text{d}=\frac{9-\sqrt{33}}{4}\)

Solve by using the Quadratic Formula: \(4{x}^{2}-20x=-25.\)

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Add 25 to get the equation in standard form.
Identify the values of a, b, and c.
Write the quadratic formula.
Then substitute in the values of a, b, and c.
Simplify.

Simplify the radical.
Simplify the fraction.
Check: We leave the check for you!

Did you recognize that 4x² − 20x + 25 is a perfect square trinomial. It is equivalent to (2x − 5)²? If you solve
4x² − 20x + 25 = 0 by factoring and then using the Square Root Property, do you get the same result?

Solve by using the Quadratic Formula: \({r}^{2}+10r+25=0.\)

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\(r=-5\)
Solve by using the Quadratic Formula: \(25{t}^{2}-40t=-16.\)

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\(t=\frac{4}{5}\)

Determine the number of solutions to each quadratic equation.

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To determine the number of solutions of each quadratic equation, we will look at its discriminant.

ⓐ

	\(3{x}^{2}+7x-9=0\)
The equation is in standard form, identify a, b, and c.	\(a=3,\ b=7,\ c=-9\)
Write the discriminant.	\({b}^{2}-4ac\)
Substitute in the values of a, b, and c.	\({(7)}^{2}-4\cdot 3\cdot (-9)\)
Simplify.	\(49+108\)
	\(157\)

Since the discriminant is positive, there are 2 real solutions to the equation.

ⓑ

	\(5{n}^{2}+n+4=0\)
The equation is in standard form, identify a, b, and c.	\(a=5,\ b=1,\ c=4\)
Write the discriminant.	\({b}^{2}-4ac\)
Substitute in the values of a, b, and c.	\({(1)}^{2}-4\cdot 5\cdot 4\)
Simplify.	\(1-80\)
	\(-79\)

Since the discriminant is negative, there are 2 complex solutions to the equation.

ⓒ

	\(9{y}^{2}-6y+1=0\)
The equation is in standard form, identify a, b, and c.	\(a=9,b=-6,c=1\)
Write the discriminant.	\({b}^{2}-4ac\)
Substitute in the values of a, b, and c.	\({(-6)}^{2}-4\cdot 9\cdot 1\)
Simplify.	\(36-36\)
	\(0\)

Since the discriminant is 0, there is 1 real solution to the equation.

Determine the numberand type of solutions to each quadratic equation.
ⓐ \(8{m}^{2}-3m+6=0\) ⓑ \(5{z}^{2}+6z-2=0\) ⓒ \(9{w}^{2}+24w+16=0.\)

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ⓐ 2 complex solutions; ⓑ 2 real solutions; ⓒ 1 real solution
Determine the number and type of solutions to each quadratic equation.
ⓐ \({b}^{2}+7b-13=0\) ⓑ \(5{a}^{2}-6a+10=0\) ⓒ \(4{r}^{2}-20r+25=0.\)

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ⓐ 2 real solutions; ⓑ 2 complex solutions; ⓒ 1 real solution
Identify the most appropriate method to use to solve each quadratic equation.
ⓐ \(5{z}^{2}=17\) ⓑ \(4{x}^{2}-12x+9=0\) ⓒ \(8{u}^{2}+6u=11.\)

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ⓐ
\(\begin{array}{l} \\ \\ \\ 5{z}^{2}=17\end{array}\)
Since the equation is in the \(a{x}^{2}=k,\) the most appropriate method is to use the Square Root Property.
ⓑ
\(\begin{array}{l} \\ \\ \\ 4{x}^{2}-12x+9=0\end{array}\)
We recognize that the left side of the equation is a perfect square trinomial, and so factoring will be the most appropriate method.
ⓒ
\(8{u}^{2}+6u=11\)
Put the equation in standard form. \(8{u}^{2}+6u-11=0\)

While our first thought may be to try factoring, thinking about all the possibilities for trial and error method leads us to choose the Quadratic Formula as the most appropriate method.
Identify the most appropriate method to use to solve each quadratic equation.
ⓐ \({x}^{2}+6x+8=0\) ⓑ \({(n-3)}^{2}=16\) ⓒ \(5{p}^{2}-6p=9.\)

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ⓐ factoring; ⓑ Square Root Property; ⓒ Quadratic Formula
Identify the most appropriate method to use to solve each quadratic equation.
ⓐ \(8{a}^{2}+3a-9=0\) ⓑ \(4{b}^{2}+4b+1=0\) ⓒ \(5{c}^{2}=125.\)

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ⓐ Quadratic Forumula;
ⓑ Factoring or Square Root Property ⓒ Square Root Property
\(4{m}^{2}+m-3=0\)

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\(m=-1,m=\frac{3}{4}\)
\(4{n}^{2}-9n+5=0\)
\(2{p}^{2}-7p+3=0\)

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\(p=\frac{1}{2},p=3\)
\(3{q}^{2}+8q-3=0\)
\({p}^{2}+7p+12=0\)

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\(p=-4,p=-3\)
\({q}^{2}+3q-18=0\)
\({r}^{2}-8r=33\)

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\(r=-3,r=11\)
\({t}^{2}+13t=-40\)
\(3{u}^{2}+7u-2=0\)

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\(u=\frac{-7\pm \sqrt{73}}{6}\)
\(2{p}^{2}+8p+5=0\)

Symbols used here

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

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

imaginary unit
i² = −1.

\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.

|x|

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

\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: Solve Quadratic Equations Using the Quadratic Formula

Solve quadratic equations using the Quadratic Formula
Use the discriminant to predict the number and type of solutions of a quadratic equation
Identify the most appropriate method to use to solve a quadratic equation
Write the quadratic equation in standard form,
Write the Quadratic Formula. Then substitute in the values of
Simplify.
Check the solutions.
If

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.

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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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