maths.free › Algebra › 9. Quadratic Equations and Functions › Solve Quadratic Equations by Completing the Square

Solve Quadratic Equations by Completing the Square

Complete the square of a binomial expression

Complete the Square of a Binomial Expression

In the last section, we were able to use the Square Root Property to solve the equation (y − 7)² = 12 because the left side was a perfect square.

\[\begin{array}{lll}{(y-7)}^{2} & = & 12 \\ y-7 & = & \pm \sqrt{12} \\ y-7 & = & \pm 2\sqrt{3} \\ y & = & 7\pm 2\sqrt{3}\end{array}\]

We also solved an equation in which the left side was a perfect square trinomial, but we had to rewrite it the form \({(x-k)}^{2}\) in order to use the Square Root Property.

\[\begin{array}{lll}{x}^{2}-10x+25 & = & 18 \\ {(x-5)}^{2} & = & 18\end{array}\]

What happens if the variable is not part of a perfect square? Can we use algebra to make a perfect square?

Let’s look at two examples to help us recognize the patterns.

\[\begin{array}{llll}{(x+9)}^{2} & & & \ {(y-7)}^{2} \\ (x+9)(x+9) & & & \ (y-7)(y-7) \\ {x}^{2}+9x+9x+81 & & & \ {y}^{2}-7y-7y+49 \\ {x}^{2}+18x+81 & & & \ {y}^{2}-14y+49\end{array}\]

We restate the patterns here for reference.

We can use this pattern to “make” a perfect square.

We will start with the expression x² + 6x. Since there is a plus sign between the two terms, we will use the (a + b)² pattern, a² + 2ab + b² = (a + b)².

Example

Try it.

Complete the square to make a perfect square trinomial. Then write the result as a binomial squared.

ⓐ \({x}^{2}-26x\) ⓑ \({y}^{2}-9y\) ⓒ \({n}^{2}+\frac{1}{2}n\)

Solution

ⓐ


The coefficient of \(x\) is −26.
\(\begin{array}{l} \\ \\ \\ \text{Find}\ {(\frac{1}{2}b)}^{2}. \\ {(\frac{1}{2}\cdot (-26))}^{2} \\ {(13)}^{2} \\ 169\end{array}\)
Add 169 to the binomial to complete the square.
Factor the perfect square trinomial, writing it as a binomial squared.

ⓑ


The coefficient of \(y\) is \(-9\).
\(\begin{array}{l} \\ \\ \\ \text{Find}\ {(\frac{1}{2}b)}^{2}. \\ {(\frac{1}{2}\cdot (-9))}^{2} \\ {(\text{-}\frac{9}{2})}^{2} \\ \frac{81}{4}\end{array}\)
Add \(\frac{81}{4}\) to the binomial to complete the square.
Factor the perfect square trinomial, writing it as a binomial squared.

ⓒ


The coefficient of \(n\) is \(\frac{1}{2}.\)
\(\begin{array}{l} \\ \\ \\ \text{Find}\ {(\frac{1}{2}b)}^{2}. \\ {(\frac{1}{2}\cdot \frac{1}{2})}^{2} \\ {(\frac{1}{4})}^{2} \\ \frac{1}{16}\end{array}\)
Add \(\frac{1}{16}\) to the binomial to complete the square.
Rewrite as a binomial square.

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

Solve Quadratic Equations of the Form

In solving equations, we must always do the same thing to both sides of the equation. This is true, of course, when we solve a quadratic equation by completing the square too. When we add a term to one side of the equation to make a perfect square trinomial, we must also add the same term to the other side of the equation.

For example, if we start with the equation x² + 6x = 40, and we want to complete the square on the left, we will add 9 to both sides of the equation.




Add 9 to both sides to complete the square.

Now the equation is in the form to solve using the Square Root Property! Completing the square is a way to transform an equation into the form we need to be able to use the Square Root Property.

How to Solve a Quadratic Equation of the Form

Try it.

Solve by completing the square: \({x}^{2}+8x=48.\)

Solution

The steps to solve a quadratic equation by completing the square are listed here.

When we solve an equation by completing the square, the answers will not always be integers.

Example

Try it.

Solve by completing the square: \({x}^{2}+4x=-21.\)

Solution


The variable terms are on the left side. Take half of 4 and square it.
\({(\frac{1}{2}(4))}^{2}=4\)
Add 4 to both sides.
Factor the perfect square trinomial, writing it as a binomial squared.
Use the Square Root Property.
Simplify using complex numbers.
Subtract 2 from each side.
Rewrite to show two solutions.
We leave the check to you.

In the previous example, our solutions were complex numbers. In the next example, the solutions will be irrational numbers.

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

Solve Quadratic Equations of the Form

The process of completing the square works best when the coefficient of x² is 1, so the left side of the equation is of the form x² + bx + c. If the x² term has a coefficient other than 1, we take some preliminary steps to make the coefficient equal to 1.

Sometimes the coefficient can be factored from all three terms of the trinomial. This will be our strategy in the next example.

Example

Try it.

Solve by completing the square: \(3{x}^{2}-12x-15=0.\)

Solution

To complete the square, we need the coefficient of \({x}^{2}\) to be one. If we factor out the coefficient of \({x}^{2}\) as a common factor, we can continue with solving the equation by completing the square.


Factor out the greatest common factor.
Divide both sides by 3 to isolate the trinomial with coefficient 1.
Simplify.
Add 5 to get the constant terms on the right side.
Take half of 4 and square it.
\({(\frac{1}{2}(-4))}^{2}=4\)
Add 4 to both sides.
Factor the perfect square trinomial, writing it as a binomial squared.
Use the Square Root Property.
Solve for x.
Rewrite to show two solutions.
Simplify.
Check:

To complete the square, the coefficient of the x² must be 1. When the leading coefficient is not a factor of all the terms, we will divide both sides of the equation by the leading coefficient! This will give us a fraction for the second coefficient. We have already seen how to complete the square with fractions in this section.

Example

Try it.

Solve by completing the square: \(2{x}^{2}-3x=20.\)

Solution

To complete the square we need the coefficient of \({x}^{2}\) to be one. We will divide both sides of the equation by the coefficient of x². Then we can continue with solving the equation by completing the square.


Divide both sides by 2 to get the coefficient of \({x}^{2}\) to be 1.
Simplify.
Take half of \(-\frac{3}{2}\) and square it.
\({(\frac{1}{2}(-\frac{3}{2}))}^{2}=\frac{9}{16}\)
Add \(\frac{9}{16}\) to both sides.
Factor the perfect square trinomial, writing it as a binomial squared.
Add the fractions on the right side.
Use the Square Root Property.
Simplify the radical.
Solve for x.
Rewrite to show two solutions.
Simplify.
Check: We leave the check for you!

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

Key Concepts

Binomial Squares Pattern
If a and b are real numbers,
How to Complete a Square
1. Identify b, the coefficient of x.
2. Find \({(\frac{1}{2}b)}^{2},\) the number to complete the square.
3. Add the \({(\frac{1}{2}b)}^{2}\) to x² + bx
4. Rewrite the trinomial as a binomial square
How to solve a quadratic equation of the form ax² + bx + c = 0 by completing the square.
1. Divide by a to make the coefficient of x² term 1.
2. Isolate the variable terms on one side and the constant terms on the other.
3. Find \({(\frac{1}{2}\ \cdot \ b)}^{2},\) the number needed to complete the square. Add it to both sides of the equation.
4. Factor the perfect square trinomial, writing it as a binomial squared on the left and simplify by adding the terms on the right.
5. Use the Square Root Property.
6. Simplify the radical and then solve the two resulting equations.
7. Check the solutions.

Solve Quadratic Equations by Completing the Square

Complete the Square of a Binomial Expression

In the following exercises, complete the square to make a perfect square trinomial. Then write the result as a binomial squared.

Try it.

ⓐ \({m}^{2}-24m\) ⓑ \({x}^{2}-11x\) ⓒ \({p}^{2}-\frac{1}{3}p\)

Solution

ⓐ \({(m-12)}^{2}\) ⓑ \({(x-\frac{11}{2})}^{2}\)
ⓒ \({(p-\frac{1}{6})}^{2}\)

Try it.

ⓐ \({n}^{2}-16n\) ⓑ \({y}^{2}+15y\) ⓒ \({q}^{2}+\frac{3}{4}q\)

Try it.

ⓐ \({p}^{2}-22p\) ⓑ \({y}^{2}+5y\) ⓒ \({m}^{2}+\frac{2}{5}m\)

Solution

ⓐ \({(p-11)}^{2}\) ⓑ \({(y+\frac{5}{2})}^{2}\)
ⓒ \({(m+\frac{1}{5})}^{2}\)

Try it.

ⓐ \({q}^{2}-6q\) ⓑ \({x}^{2}-7x\) ⓒ \({n}^{2}-\frac{2}{3}n\)

Solve Quadratic Equations of the form x² + bx + c = 0 by Completing the Square

In the following exercises, solve by completing the square.

Try it.

\({u}^{2}+2u=3\)

Solution

\(u=-3,u=1\)

Try it.

\({z}^{2}+12z=-11\)

Try it.

\({x}^{2}-20x=21\)

Solution

\(x=-1,x=21\)

Try it.

\({y}^{2}-2y=8\)

Try it.

\({m}^{2}+4m=-44\)

Solution

\(m=-2\pm 2\sqrt{10}i\)

Try it.

\({n}^{2}-2n=-3\)

Try it.

\({r}^{2}+6r=-11\)

Solution

\(r=-3\pm \sqrt{2}i\)

Try it.

\({t}^{2}-14t=-50\)

Try it.

\({a}^{2}-10a=-5\)

Solution

\(a=5\pm 2\sqrt{5}\)

Try it.

\({b}^{2}+6b=41\)

Try it.

\({x}^{2}+5x=2\)

Solution

\(x=-\frac{5}{2}\pm \frac{\sqrt{33}}{2}\)

Try it.

\({y}^{2}-3y=2\)

Try it.

\({u}^{2}-14u+12=-1\)

Solution

\(u=1,u=13\)

Try it.

\({z}^{2}+2z-5=2\)

Try it.

\({r}^{2}-4r-3=9\)

Solution

\(r=-2,r=6\)

Try it.

\({t}^{2}-10t-6=5\)

Try it.

\({v}^{2}=9v+2\)

Solution

\(v=\frac{9}{2}\pm \frac{\sqrt{89}}{2}\)

Try it.

\({w}^{2}=5w-1\)

Try it.

\({x}^{2}-5=10x\)

Solution

\(x=5\pm \sqrt{30}\)

Try it.

\({y}^{2}-14=6y\)

Try it.

\((x+6)(x-2)=9\)

Solution

\(x=-7,x=3\)

Try it.

\((y+9)(y+7)=80\)

Try it.

\((x+2)(x+4)=3\)

Solution

\(x=-5,x=-1\)

Try it.

\((x-2)(x-6)=5\)

Solve Quadratic Equations of the form ax² + bx + c = 0 by Completing the Square

In the following exercises, solve by completing the square.

Try it.

\(3{m}^{2}+30m-27=6\)

Solution

\(m=-11,m=1\)

Try it.

\(2{x}^{2}-14x+12=0\)

Try it.

\(2{n}^{2}+4n=26\)

Solution

\(n=-1\pm \sqrt{14}\)

Try it.

\(5{x}^{2}+20x=15\)

Try it.

\(2{c}^{2}+c=6\)

Solution

\(c=-2,c=\frac{3}{2}\)

Try it.

\(3{d}^{2}-4d=15\)

Try it.

\(2{x}^{2}+7x-15=0\)

Solution

\(x=-5,x=\frac{3}{2}\)

Try it.

\(3{x}^{2}-14x+8=0\)

Try it.

\(2{p}^{2}+7p=14\)

Solution

\(p=-\frac{7}{4}\pm \frac{\sqrt{161}}{4}\)

Try it.

\(3{q}^{2}-5q=9\)

Try it.

\(5{x}^{2}-3x=-10\)

Solution

\(x=\frac{3}{10}\pm \frac{\sqrt{191}}{10}i\)

Try it.

\(7{x}^{2}+4x=-3\)

Try it.

Solve the equation \({x}^{2}+10x=-25\)

ⓐ by using the Square Root Property

ⓑ by Completing the Square

Solution

Answers will vary.

Try it.

Solve the equation \({y}^{2}+8y=48\) by completing the square and explain all your steps.

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?

Complete The Square of a Binomial Expression

In the last section, we were able to use the Square Root Property to solve the equation \({(y-7)}^{2}=12\) because the left side was a perfect square.

\[\begin{array}{lll}{(y-7)}^{2} & = & 12 \\ y-7 & = & \pm \ \sqrt{12} \\ y-7 & = & \pm \ 2\sqrt{3} \\ y & = & 7\pm 2\sqrt{3}\end{array}\]

We also solved an equation in which the left side was a perfect square trinomial, but we had to rewrite it the form \({(x-k)}^{2}\) in order to use the square root property.

\[\begin{array}{lll}{x}^{2}-10x+25 & = & 18 \\ {(x-5)}^{2} & = & 18\end{array}\]

What happens if the variable is not part of a perfect square? Can we use algebra to make a perfect square?

Let’s study the binomial square pattern we have used many times. We will look at two examples.

\[\begin{array}{llll}\begin{array}{l}{(x+9)}^{2} \\ (x+9)(x+9) \\ {x}^{2}+9x+9x+81 \\ {x}^{2}+18x+81\end{array} & & & \ \begin{array}{l}{(y-7)}^{2} \\ (y-7)\ (y-7) \\ {y}^{2}-7y-7y+49 \\ {y}^{2}-14y+49\end{array}\end{array}\]

We can use this pattern to “make” a perfect square.

We will start with the expression \({x}^{2}+6x\). Since there is a plus sign between the two terms, we will use the \({(a+b)}^{2}\) pattern.

\[{a}^{2}+2ab+{b}^{2}={(a+b)}^{2}\]

Notice that the first term of \({x}^{2}+6x\) is a square, \({x}^{2}\).

Example

Try it.

Complete the square to make a perfect square trinomial. Then, write the result as a binomial square.

\({x}^{2}+14x\)

Solution

The coefficient of x is 14.
\(\begin{array}{l} \\ \\ \\ \text{Find}\ {(\frac{1}{2}b)}^{2}. \\ {(\frac{1}{2}⋅14)}^{2} \\ {(7)}^{2} \\ 49\end{array}\)
Add 49 to the binomial to complete the square.	\({x}^{2}+14x+49\)
Rewrite as a binomial square.	\({(x+7)}^{2}\)

Example

Try it.

Complete the square to make a perfect square trinomial. Then, write the result as a binomial squared. \({m}^{2}-26m\)

Solution

The coefficient of m is −26.
\(\begin{array}{l} \\ \\ \\ \text{Find}\ {(\frac{1}{2}b)}^{2}. \\ {(\frac{1}{2}⋅(\text{-}26))}^{2} \\ {(\text{-}13)}^{2} \\ 169\end{array}\)
Add 169 to the binomial to complete the square.	\({m}^{2}-26m+169\)
Rewrite as a binomial square.	\({(m-13)}^{2}\)

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

Solve Quadratic Equations of the Form

In solving equations, we must always do the same thing to both sides of the equation. This is true, of course, when we solve a quadratic equation by completing the square, too. When we add a term to one side of the equation to make a perfect square trinomial, we must also add the same term to the other side of the equation.

For example, if we start with the equation \({x}^{2}+6x=40\) and we want to complete the square on the left, we will add nine to both sides of the equation.

Then, we factor on the left and simplify on the right.

\[{(x+3)}^{2}=49\]

Now the equation is in the form to solve using the Square Root Property. Completing the square is a way to transform an equation into the form we need to be able to use the Square Root Property.

How To Solve a Quadratic Equation of the Form

Try it.

Solve \({x}^{2}+8x=48\) by completing the square.

Solution

Example

Try it.

Solve \({y}^{2}-6y=16\) by completing the square.

Solution

The variable terms are on the left side.
Take half of \(-6\) and square it. \((\frac{1}{2}(\text{-}6){)}^{2}=9\)
Add 9 to both sides.
Factor the perfect square trinomial as a binomial square.
Use the Square Root Property.
Simplify the radical.
Solve for y.
Rewrite to show two solutions.
Solve the equations.
Check.

Example

Try it.

Solve \({x}^{2}+4x=-21\) by completing the square.

Solution

The variable terms are on the left side.
Take half of \(4\) and square it. \((\frac{1}{2}(4){)}^{2}=4\)
Add 4 to both sides.
Factor the perfect square trinomial as a binomial square.
Use the Square Root Property.
We cannot take the square root of a negative number.	There is no real solution.

In the previous example, there was no real solution because \({(x+k)}^{2}\) was equal to a negative number.

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

Solve Quadratic Equations of the form

The process of completing the square works best when the leading coefficient is one, so the left side of the equation is of the form \({x}^{2}+bx+c\). If the \({x}^{2}\) term has a coefficient, we take some preliminary steps to make the coefficient equal to one.

Sometimes the coefficient can be factored from all three terms of the trinomial. This will be our strategy in the next example.

Example

Try it.

Solve \(3{x}^{2}-12x-15=0\) by completing the square.

Solution


Factor out the greatest common factor.
Divide both sides by 3 to isolate the trinomial.
Simplify.
Subtract 5 to get the constant terms on the right.
Take half of 4 and square it. \((\frac{1}{2}(4){)}^{2}=4\)
Add 4 to both sides.
Factor the perfect square trinomial as a binomial square.
Use the Square Root Property.
Solve for x.
Rewrite to show 2 solutions.
Simplify.
Check.

To complete the square, the leading coefficient must be one. When the leading coefficient is not a factor of all the terms, we will divide both sides of the equation by the leading coefficient. This will give us a fraction for the second coefficient. We have already seen how to complete the square with fractions in this section.

Example

Try it.

Solve \(2{x}^{2}-3x=20\) by completing the square.

Solution

Again, our first step will be to make the coefficient of \({x}^{2}\) be one. By dividing both sides of the equation by the coefficient of \({x}^{2}\), we can then continue with solving the equation by completing the square.


Divide both sides by 2 to get the coefficient of \({x}^{2}\) to be 1.
Simplify.
Take half of \(-\frac{3}{2}\) and square it. \((\frac{1}{2}(-\frac{3}{2}){)}^{2}=\frac{9}{16}\)
Add \(\frac{9}{16}\) to both sides.
Factor the perfect square trinomial as a binomial square.
Add the fractions on the right side.
Use the Square Root Property.
Simplify the radical.
Solve for x.
Rewrite to show 2 solutions.
Simplify.
Check. We leave the check for you.

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

Key Concepts

Binomial Squares Pattern If \(a,b\) are real numbers,
\({(a+b)}^{2}={a}^{2}+2ab+{b}^{2}\)

\({(a-b)}^{2}={a}^{2}-2ab+{b}^{2}\)
Complete a Square
To complete the square of \({x}^{2}+bx\):
1. Identify \(b\), the coefficient of \(x\).
2. Find \({(\frac{1}{2}b)}^{2}\), the number to complete the square.
3. Add the \({(\frac{1}{2}b)}^{2}\) to \({x}^{2}+bx\).

Solve Quadratic Equations by Completing the Square

Complete the Square of a Binomial Expression

In the following exercises, complete the square to make a perfect square trinomial. Then, write the result as a binomial squared.

Try it.

\({a}^{2}+10a\)

Solution

\({(a+5)}^{2}\)

Try it.

\({b}^{2}+12b\)

Try it.

\({m}^{2}+18m\)

Solution

\({(m+9)}^{2}\)

Try it.

\({n}^{2}+16n\)

Try it.

\({m}^{2}-24m\)

Solution

\({(m-12)}^{2}\)

Try it.

\({n}^{2}-16n\)

Try it.

\({p}^{2}-22p\)

Solution

\({(p-11)}^{2}\)

Try it.

\({q}^{2}-6q\)

Try it.

\({x}^{2}-9x\)

Solution

\({(x-\frac{9}{2})}^{2}\)

Try it.

\({y}^{2}+11y\)

Try it.

\({p}^{2}-\frac{1}{3}p\)

Solution

\({(p-\frac{1}{6})}^{2}\)

Try it.

\({q}^{2}+\frac{3}{4}q\)

Solve Quadratic Equations of the Form \({x}^{2}+bx+c=0\) by Completing the Square

In the following exercises, solve by completing the square.

Try it.

\({v}^{2}+6v=40\)

Solution

\(v=-10,v=4\)

Try it.

\({w}^{2}+8w=65\)

Try it.

\({u}^{2}+2u=3\)

Solution

\(u=-3,u=1\)

Try it.

\({z}^{2}+12z=-11\)

Try it.

\({c}^{2}-12c=13\)

Solution

\(c=-1,c=13\)

Try it.

\({d}^{2}-8d=9\)

Try it.

\({x}^{2}-20x=21\)

Solution

\(x=-1,x=21\)

Try it.

\({y}^{2}-2y=8\)

Try it.

\({m}^{2}+4m=-44\)

Solution

no real solution

Try it.

\({n}^{2}-2n=-3\)

Try it.

\({r}^{2}+6r=-11\)

Solution

no real solution

Try it.

\({t}^{2}-14t=-50\)

Try it.

\({a}^{2}-10a=-5\)

Solution

\(a=5\pm 2\sqrt{5}\)

Try it.

\({b}^{2}+6b=41\)

Try it.

\({u}^{2}-14u+12=-1\)

Solution

\(u=1,u=13\)

Try it.

\({z}^{2}+2z-5=2\)

Try it.

\({v}^{2}=9v+2\)

Solution

\(v=\frac{9}{2}\pm \frac{\sqrt{89}}{2}\)

Try it.

\({w}^{2}=5w-1\)

Try it.

\((x+6)(x-2)=9\)

Solution

\(x=-7,x=3\)

Try it.

\((y+9)(y+7)=79\)

Solve Quadratic Equations of the Form \(a{x}^{2}+bx+c=0\) by Completing the Square

In the following exercises, solve by completing the square.

Try it.

\(3{m}^{2}+30m-27=6\)

Solution

\(m=-11,m=1\)

Try it.

\(2{n}^{2}+4n-26=0\)

Try it.

\(2{c}^{2}+c=6\)

Solution

\(c=-2,c=\frac{3}{2}\)

Try it.

\(3{d}^{2}-4d=15\)

Try it.

\(2{p}^{2}+7p=14\)

Solution

\(p=-\frac{7}{4}\pm \frac{\sqrt{161}}{4}\)

Try it.

\(3{q}^{2}-5q=9\)

Try it.

Rafi is designing a rectangular playground to have an area of 320 square feet. He wants one side of the playground to be four feet longer than the other side. Solve the equation \({p}^{2}+4p=320\) for \(p\), the length of one side of the playground. What is the length of the other side?

Solution

16 feet, 20 feet

Try it.

Yvette wants to put a square swimming pool in the corner of her backyard. She will have a 3 foot deck on the south side of the pool and a 9 foot deck on the west side of the pool. She has a total area of 1080 square feet for the pool and two decks. Solve the equation \((s+3)(s+9)=1080\) for \(s\), the length of a side of the pool.

Try it.

Solve the equation \({x}^{2}+10x=-25\) ⓐ by using the Square Root Property and ⓑ by completing the square. ⓒ Which method do you prefer? Why?

Solution

Try it.

Solve the equation \({y}^{2}+8y=48\) by completing the square and explain all your steps.

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.

Expand: \({(x+9)}^{2}.\)
If you missed this problem, review .

Tunjukkan jawapan

\({x}^{2}+18x+81\)
Factor \({y}^{2}-14y+49.\)
If you missed this problem, review .

Tunjukkan jawapan

\((y-7{)}^{2}\)
Factor \(5{n}^{2}+40n+80.\)
If you missed this problem, review .

Tunjukkan jawapan

\(5(n+4{)}^{2}\)

Complete the square to make a perfect square trinomial. Then write the result as a binomial squared.

Tunjukkan jawapan

ⓐ


The coefficient of \(x\) is −26.
\(\begin{array}{l} \\ \\ \\ \text{Find}\ {(\frac{1}{2}b)}^{2}. \\ {(\frac{1}{2}\cdot (-26))}^{2} \\ {(13)}^{2} \\ 169\end{array}\)
Add 169 to the binomial to complete the square.
Factor the perfect square trinomial, writing it as a binomial squared.

ⓑ


The coefficient of \(y\) is \(-9\).
\(\begin{array}{l} \\ \\ \\ \text{Find}\ {(\frac{1}{2}b)}^{2}. \\ {(\frac{1}{2}\cdot (-9))}^{2} \\ {(\text{-}\frac{9}{2})}^{2} \\ \frac{81}{4}\end{array}\)
Add \(\frac{81}{4}\) to the binomial to complete the square.
Factor the perfect square trinomial, writing it as a binomial squared.

ⓒ


The coefficient of \(n\) is \(\frac{1}{2}.\)
\(\begin{array}{l} \\ \\ \\ \text{Find}\ {(\frac{1}{2}b)}^{2}. \\ {(\frac{1}{2}\cdot \frac{1}{2})}^{2} \\ {(\frac{1}{4})}^{2} \\ \frac{1}{16}\end{array}\)
Add \(\frac{1}{16}\) to the binomial to complete the square.
Rewrite as a binomial square.

Complete the square to make a perfect square trinomial. Then write the result as a binomial squared.
ⓐ \({a}^{2}-20a\) ⓑ \({m}^{2}-5m\) ⓒ \({p}^{2}+\frac{1}{4}p\)

Tunjukkan jawapan

ⓐ \({(a-10)}^{2}\) ⓑ \({(b-\frac{5}{2})}^{2}\)
ⓒ \({(p+\frac{1}{8})}^{2}\)
Complete the square to make a perfect square trinomial. Then write the result as a binomial squared.
ⓐ \({b}^{2}-4b\) ⓑ \({n}^{2}+13n\) ⓒ \({q}^{2}-\frac{2}{3}q\)

Tunjukkan jawapan

ⓐ \({(b-2)}^{2}\) ⓑ \({(n+\frac{13}{2})}^{2}\)
ⓒ \({(q-\frac{1}{3})}^{2}\)
Solve by completing the square: \({x}^{2}+8x=48.\)
Solve by completing the square: \({x}^{2}+4x=5.\)

Tunjukkan jawapan

\(x=-5,x=1\)
Solve by completing the square: \({y}^{2}-10y=-9.\)

Tunjukkan jawapan

\(y=1,y=9\)

Solve by completing the square: \({x}^{2}+4x=-21.\)

Tunjukkan jawapan


The variable terms are on the left side. Take half of 4 and square it.
\({(\frac{1}{2}(4))}^{2}=4\)
Add 4 to both sides.
Factor the perfect square trinomial, writing it as a binomial squared.
Use the Square Root Property.
Simplify using complex numbers.
Subtract 2 from each side.
Rewrite to show two solutions.
We leave the check to you.

Solve by completing the square: \({y}^{2}-10y=-35.\)

Tunjukkan jawapan

\(y=5\pm \sqrt{10}i\)
Solve by completing the square: \({z}^{2}+8z=-19.\)

Tunjukkan jawapan

\(z=-4+\sqrt{3}i,\ \text{z}=-4-\sqrt{3}i\)

Solve by completing the square: \({y}^{2}-18y=-6.\)

Tunjukkan jawapan


The variable terms are on the left side. Take half of \(-18\) and square it.
\({(\frac{1}{2}(-18))}^{2}=81\)
Add 81 to both sides.
Factor the perfect square trinomial, writing it as a binomial squared.
Use the Square Root Property.
Simplify the radical.
Solve for \(y\).
Check.

Another way to check this would be to use a calculator. Evaluate \({y}^{2}-18y\)for both of the solutions. The answer should be \(-6.\)

Solve by completing the square: \({x}^{2}-16x=-16.\)

Tunjukkan jawapan

\(x=8+4\sqrt{3},\ \ x=8-4\sqrt{3}\)
Solve by completing the square: \({y}^{2}+8y=11.\)

Tunjukkan jawapan

\(y=-4+3\sqrt{3},\ \text{y}=-4-3\sqrt{3}\)

Solve by completing the square: \({x}^{2}+10x+4=15.\)

Tunjukkan jawapan


Isolate the variable terms on the left side. Subtract 4 to get the constant terms on the right side.
Take half of 10 and square it.
\({(\frac{1}{2}(10))}^{2}=25\)
Add 25 to both sides.
Factor the perfect square trinomial, writing it as a binomial squared.
Use the Square Root Property.
Simplify the radical.
Solve for x.
Rewrite to show two solutions.
Solve the equations.
Check:

Solve by completing the square: \({a}^{2}+4a+9=30.\)

Tunjukkan jawapan

\(a=-7,a=3\)
Solve by completing the square: \({b}^{2}+8b-4=16.\)

Tunjukkan jawapan

\(b=-10,b=2\)

Solve by completing the square: \({n}^{2}=3n+11.\)

Tunjukkan jawapan


Subtract \(3n\) to get the variable terms on the left side.
Take half of \(-3\) and square it.
\({(\frac{1}{2}(-3))}^{2}=\frac{9}{4}\)
Add \(\frac{9}{4}\) to both sides.
Factor the perfect square trinomial, writing it as a binomial squared.
Add the fractions on the right side.
Use the Square Root Property.
Simplify the radical.
Solve for n.
Rewrite to show two solutions.
Check: We leave the check for you!

Solve by completing the square: \({p}^{2}=5p+9.\)

Tunjukkan jawapan

\(p=\frac{5}{2}+\frac{\sqrt{61}}{2},\ \ p=\frac{5}{2}-\frac{\sqrt{61}}{2}\)
Solve by completing the square: \({q}^{2}=7q-3.\)

Tunjukkan jawapan

\(q=\frac{7}{2}+\frac{\sqrt{37}}{2},\ \text{q}=\frac{7}{2}-\frac{\sqrt{37}}{2}\)

Solve by completing the square: \((x-3)(x+5)=9.\)

Tunjukkan jawapan


We multiply the binomials on the left.
Add 15 to isolate the constant terms on the right.
Take half of 2 and square it.
\({(\frac{1}{2}\cdot (2))}^{2}=1\)
Add 1 to both sides.
Factor the perfect square trinomial, writing it as a binomial squared.
Use the Square Root Property.
Solve for x.
Rewrite to show two solutions.
Simplify.
Check: We leave the check for you!

Solve by completing the square: \((c-2)(c+8)=11.\)

Tunjukkan jawapan

\(c=-9,c=3\)
Solve by completing the square: \((d-7)(d+3)=56.\)

Tunjukkan jawapan

\(d=11,d=-7\)

Solve by completing the square: \(3{x}^{2}-12x-15=0.\)

Tunjukkan jawapan


Factor out the greatest common factor.
Divide both sides by 3 to isolate the trinomial with coefficient 1.
Simplify.
Add 5 to get the constant terms on the right side.
Take half of 4 and square it.
\({(\frac{1}{2}(-4))}^{2}=4\)
Add 4 to both sides.
Factor the perfect square trinomial, writing it as a binomial squared.
Use the Square Root Property.
Solve for x.
Rewrite to show two solutions.
Simplify.
Check:

Solve by completing the square: \(2{m}^{2}+16m+14=0.\)

Tunjukkan jawapan

\(m=-7,m=-1\)
Solve by completing the square: \(4{n}^{2}-24n-56=8.\)

Tunjukkan jawapan

\(n=-2,n=8\)

Solve by completing the square: \(2{x}^{2}-3x=20.\)

Tunjukkan jawapan


Divide both sides by 2 to get the coefficient of \({x}^{2}\) to be 1.
Simplify.
Take half of \(-\frac{3}{2}\) and square it.
\({(\frac{1}{2}(-\frac{3}{2}))}^{2}=\frac{9}{16}\)
Add \(\frac{9}{16}\) to both sides.
Factor the perfect square trinomial, writing it as a binomial squared.
Add the fractions on the right side.
Use the Square Root Property.
Simplify the radical.
Solve for x.
Rewrite to show two solutions.
Simplify.
Check: We leave the check for you!

Solve by completing the square: \(3{r}^{2}-2r=21.\)

Tunjukkan jawapan

\(r=-\frac{7}{3},r=3\)
Solve by completing the square: \(4{t}^{2}+2t=20.\)

Tunjukkan jawapan

\(t=-\frac{5}{2},t=2\)

Solve by completing the square: \(3{x}^{2}+2x=4.\)

Tunjukkan jawapan

Again, our first step will be to make the coefficient of x² one. By dividing both sides of the equation by the coefficient of x², we can then continue with solving the equation by completing the square.


Divide both sides by 3 to make the coefficient of \({x}^{2}\) equal 1.
Simplify.
Take half of \(\frac{2}{3}\) and square it.
\({(\frac{1}{2}\cdot \frac{2}{3})}^{2}=\frac{1}{9}\)
Add \(\frac{1}{9}\) to both sides.
Factor the perfect square trinomial, writing it as a binomial squared.
Use the Square Root Property.
Simplify the radical.
Solve for x .
Rewrite to show two solutions.
Check: We leave the check for you!

Solve by completing the square: \(4{x}^{2}+3x=2.\)

Tunjukkan jawapan

\(x=-\frac{3}{8}+\frac{\sqrt{41}}{8},\ \ x=-\frac{3}{8}-\frac{\sqrt{41}}{8}\)
Solve by completing the square: \(3{y}^{2}-10y=-5.\)

Tunjukkan jawapan

\(y=\frac{5}{3}+\frac{\sqrt{10}}{3},\ \ y=\frac{5}{3}-\frac{\sqrt{10}}{3}\)
ⓐ \({m}^{2}-24m\) ⓑ \({x}^{2}-11x\) ⓒ \({p}^{2}-\frac{1}{3}p\)

Tunjukkan jawapan

ⓐ \({(m-12)}^{2}\) ⓑ \({(x-\frac{11}{2})}^{2}\)
ⓒ \({(p-\frac{1}{6})}^{2}\)
ⓐ \({p}^{2}-22p\) ⓑ \({y}^{2}+5y\) ⓒ \({m}^{2}+\frac{2}{5}m\)

Tunjukkan jawapan

ⓐ \({(p-11)}^{2}\) ⓑ \({(y+\frac{5}{2})}^{2}\)
ⓒ \({(m+\frac{1}{5})}^{2}\)
\({u}^{2}+2u=3\)

Tunjukkan jawapan

\(u=-3,u=1\)
\({z}^{2}+12z=-11\)
\({x}^{2}-20x=21\)

Tunjukkan jawapan

\(x=-1,x=21\)

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 by Completing the Square

Complete the square of a binomial expression
Solve quadratic equations of the form
Solve quadratic equations of the form
Identify
Find
Add the
Factor the perfect square trinomial, writing it as a binomial squared.
Isolate the variable terms on one side and the constant terms on the other.

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.

Cubalah sendiri

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.

Lebih dalam Algebra

Linear equations Quadratic equations Systems of equations Inequalities Factoring Expanding Simplifying expressions Functions and graphs Exponential and logarithmic equations Polynomial equations Absolute value