Learning Objectives
By the end of this section, you will be able to:
- Complete the square of a binomial expression
- Solve quadratic equations of the form ${x}^{2}+bx+c=0$ by completing the square
- Solve quadratic equations of the form $a{x}^{2}+bx+c=0$ by completing the square
Be Prepared 10.4
Before you get started, take this readiness quiz. If you miss a problem, go back to the section listed and review the material.
Simplify ${\left(x+12\right)}^{2}$.
If you missed this problem, review Example 6.47.
Be Prepared 10.5
Factor ${y}^{2}-18y+81$.
If you missed this problem, review Example 7.42.
Be Prepared 10.6
Factor $5{n}^{2}+40n+80$.
If you missed this problem, review Example 7.46.
So far, we have solved quadratic equations by factoring and using the Square Root Property. In this section, we will solve quadratic equations by a process called ‘completing the square.’
Complete The Square of a Binomial Expression
In the last section, we were able to use the Square Root Property to solve the equation ${\left(y-7\right)}^{2}=12$ because the left side was a perfect square.
We also solved an equation in which the left side was a perfect square trinomial, but we had to rewrite it the form ${\left(x-k\right)}^{2}$ in order to use the square root property.
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.
Binomial Squares Pattern
If $a,b$ are real numbers,
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 ${\left(a+b\right)}^{2}$ pattern.
Notice that the first term of ${x}^{2}+6x$ is a square, ${x}^{2}$.
We now know $a=x$.
What number can we add to ${x}^{2}+6x$ to make a perfect square trinomial?
The middle term of the Binomial Squares Pattern, $2ab$, is twice the product of the two terms of the binomial. This means twice the product of $x$ and some number is $6x$. So, two times some number must be six. The number we need is $\frac{1}{2}\xb76=3.$ The second term in the binomial, $b,$ must be 3.
We now know $b=3$.
Now, we just square the second term of the binomial to get the last term of the perfect square trinomial, so we square three to get the last term, nine.
We can now factor to
So, we found that adding nine to ${x}^{2}+6x$ ‘completes the square,’ and we write it as ${\left(x+3\right)}^{2}$.
How To
Complete a square.
To complete the square of ${x}^{2}+bx$:
- Step 1. Identify $b$, the coefficient of $x$.
- Step 2. Find ${\left(\frac{1}{2}b\right)}^{2}$, the number to complete the square.
- Step 3. Add the ${\left(\frac{1}{2}b\right)}^{2}$ to ${x}^{2}+bx$.
Example 10.14
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. | |
$\begin{array}{}\\ \\ \\ \hfill \text{Find}\phantom{\rule{0.2em}{0ex}}{\left(\frac{1}{2}b\right)}^{2}.\hfill \\ \hfill {(\frac{1}{2}\cdot 14)}^{2}\hfill \\ \hfill {\left(7\right)}^{2}\hfill \\ \hfill 49\hfill \end{array}$ | |
Add 49 to the binomial to complete the square. | ${x}^{2}+14x+49$ |
Rewrite as a binomial square. | ${(x+7)}^{2}$ |
Try It 10.27
Complete the square to make a perfect square trinomial. Write the result as a binomial square.
${y}^{2}+12y$
Try It 10.28
Complete the square to make a perfect square trinomial. Write the result as a binomial square.
${z}^{2}+8z$
Example 10.15
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. | |
$\begin{array}{}\\ \\ \\ \hfill \text{Find}\phantom{\rule{0.2em}{0ex}}{\left(\frac{1}{2}b\right)}^{2}.\hfill \\ \hfill {(\frac{1}{2}\cdot (\text{\u2212}26\left)\right)}^{2}\hfill \\ \hfill {\left(\text{\u2212}13\right)}^{2}\hfill \\ \hfill 169\hfill \end{array}$ | |
Add 169 to the binomial to complete the square. | ${m}^{2}-26m+169$ |
Rewrite as a binomial square. | ${(m-13)}^{2}$ |
Try It 10.29
Complete the square to make a perfect square trinomial. Write the result as a binomial square.
${a}^{2}-20a$
Try It 10.30
Complete the square to make a perfect square trinomial. Write the result as a binomial square.
${b}^{2}-4b$
Example 10.16
Complete the square to make a perfect square trinomial. Then, write the result as a binomial squared.
${u}^{2}-9u$
Solution
The coefficient of u is −9. | |
$\begin{array}{}\\ \\ \\ \hfill \text{Find}\phantom{\rule{0.2em}{0ex}}{\left(\frac{1}{2}b\right)}^{2}.\hfill \\ \hfill {(\frac{1}{2}\cdot (\text{\u2212}9\left)\right)}^{2}\hfill \\ \hfill {(-\frac{9}{2})}^{2}\hfill \\ \hfill \frac{81}{4}\hfill \end{array}$ | |
Add $\frac{81}{4}$ to the binomial to complete the square. | ${u}^{2}-9u+\frac{81}{4}$ |
Rewrite as a binomial square. | ${(u-\frac{9}{2})}^{2}$ |
Try It 10.31
Complete the square to make a perfect square trinomial. Write the result as a binomial square.
${m}^{2}-5m$
Try It 10.32
Complete the square to make a perfect square trinomial. Write the result as a binomial square.
${n}^{2}+13n$
Example 10.17
Complete the square to make a perfect square trinomial. Then, write the result as a binomial squared.
${p}^{2}+\frac{1}{2}p$
Solution
The coefficient of p is $\frac{1}{2}$. | |
$\begin{array}{}\\ \\ \\ \hfill \text{Find}\phantom{\rule{0.2em}{0ex}}{\left(\frac{1}{2}b\right)}^{2}.\hfill \\ \hfill {(\frac{1}{2}\cdot \frac{1}{2})}^{2}\hfill \\ \hfill {\left(\frac{1}{4}\right)}^{2}\hfill \\ \hfill \frac{1}{16}\hfill \end{array}$ | |
Add $\frac{1}{16}$ to the binomial to complete the square. | ${p}^{2}+\frac{1}{2}p+\frac{1}{16}$ |
Rewrite as a binomial square. | ${(p+\frac{1}{4})}^{2}$ |
Try It 10.33
Complete the square to make a perfect square trinomial. Write the result as a binomial square.
${p}^{2}+\frac{1}{4}p$
Try It 10.34
Complete the square to make a perfect square trinomial. Write the result as a binomial square.
${q}^{2}-\frac{2}{3}q$
Solve Quadratic Equations of the Form x^{2} + bx + c = 0 by completing the square
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.
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.
Example 10.18
How To Solve a Quadratic Equation of the Form ${x}^{2}+bx+c=0$ by Completing the Square
Solve ${x}^{2}+8x=48$ by completing the square.
Solution
Try It 10.35
Solve ${c}^{2}+4c=5$ by completing the square.
Try It 10.36
Solve ${d}^{2}+10d=\mathrm{-9}$ by completing the square.
How To
Solve a quadratic equation of the form ${x}^{2}+bx+c=0$ by completing the square.
- Step 1. Isolate the variable terms on one side and the constant terms on the other.
- Step 2. Find ${\left(\frac{1}{2}\xb7b\right)}^{2}$, the number to complete the square. Add it to both sides of the equation.
- Step 3. Factor the perfect square trinomial as a binomial square.
- Step 4. Use the Square Root Property.
- Step 5. Simplify the radical and then solve the two resulting equations.
- Step 6. Check the solutions.
Example 10.19
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. $\left(\frac{1}{2}\right(\text{\u2212}6){)}^{2}=9$ | ||
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. |
Try It 10.37
Solve ${r}^{2}-4r=12$ by completing the square.
Try It 10.38
Solve ${t}^{2}-10t=11$ by completing the square.
Example 10.20
Solve ${x}^{2}+4x=\mathrm{-21}$ by completing the square.
Solution
The variable terms are on the left side. | ||
Take half of $4$ and square it. $\left(\frac{1}{2}\right(4){)}^{2}=4$ | ||
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. |
Try It 10.39
Solve ${y}^{2}-10y=\mathrm{-35}$ by completing the square.
Try It 10.40
Solve ${z}^{2}+8z=\mathrm{-19}$ by completing the square.
In the previous example, there was no real solution because ${\left(x+k\right)}^{2}$ was equal to a negative number.
Example 10.21
Solve ${p}^{2}-18p=\mathrm{-6}$ by completing the square.
Solution
The variable terms are on the left side. | ||
Take half of $\text{\u2212}18$ and square it. $\left(\frac{1}{2}\right(\text{\u2212}18){)}^{2}=81$ | ||
Add 81 to both sides. | ||
Factor the perfect square trinomial as a binomial square. | ||
Use the Square Root Property. | ||
Simplify the radical. | ||
Solve for p. | ||
Rewrite to show two solutions. | ||
Check. |
Another way to check this would be to use a calculator. Evaluate ${p}^{2}-18p$ for both of the solutions. The answer should be $\mathrm{-6}$.
Try It 10.41
Solve ${x}^{2}-16x=\mathrm{-16}$ by completing the square.
Try It 10.42
Solve ${y}^{2}+8y=11$ by completing the square.
We will start the next example by isolating the variable terms on the left side of the equation.
Example 10.22
Solve ${x}^{2}+10x+4=15$ by completing the square.
Solution
The variable terms are on the left side. | ||
Subtract $4$ to get the constant terms on the right side. | ||
Take half of 10 and square it. $\left(\frac{1}{2}\right(10){)}^{2}=25$ | ||
Add 25 to both sides. | ||
Factor the perfect square trinomial as a binomial square. | ||
Use the Square Root Property. | ||
Simplify the radical. | ||
Solve for x. | ||
Rewrite to show two equations. | ||
Solve the equations. | ||
Check. |
Try It 10.43
Solve ${a}^{2}+4a+9=30$ by completing the square.
Try It 10.44
Solve ${b}^{2}+8b-4=16$ by completing the square.
To solve the next equation, we must first collect all the variable terms to the left side of the equation. Then, we proceed as we did in the previous examples.
Example 10.23
Solve ${n}^{2}=3n+11$ by completing the square.
Solution
Subtract 3n to get the variable terms on the left side. | ||
Take half of $\text{\u2212}3$ and square it. $\left(\frac{1}{2}\right(\text{\u2212}3){)}^{2}=\frac{9}{4}$ | ||
Add $\frac{9}{4}$ 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 n. | ||
Rewrite to show two equations. | ||
Check. We leave the check for you! |
Try It 10.45
Solve ${p}^{2}=5p+9$ by completing the square.
Try It 10.46
Solve ${q}^{2}=7q-3$ by completing the square.
Notice that the left side of the next equation is in factored form. But the right side is not zero, so we cannot use the Zero Product Property. Instead, we multiply the factors and then put the equation into the standard form to solve by completing the square.
Example 10.24
Solve $\left(x-3\right)\left(x+5\right)=9$ by completing the square.
Solution
We multiply binomials on the left. | ||
Add 15 to get the variable terms on the left side. | ||
Take half of 2 and square it. $\left(\frac{1}{2}\right(2){)}^{2}=1$ | ||
Add 1 to both sides. | ||
Factor the perfect square trinomial as a binomial square. | ||
Use the Square Root Property. | ||
Solve for x. | ||
Rewite to show two solutions. | ||
Simplify. | ||
Check. We leave the check for you! |
Try It 10.47
Solve $\left(c-2\right)\left(c+8\right)=7$ by completing the square.
Try It 10.48
Solve $\left(d-7\right)\left(d+3\right)=56$ by completing the square.
Solve Quadratic Equations of the form ax^{2} + bx + c = 0 by completing the square
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 10.25
Solve $3{x}^{2}-12x-15=0$ by completing the square.
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. | ||
Simplify. | ||
Subtract 5 to get the constant terms on the right. | ||
Take half of 4 and square it. $\left(\frac{1}{2}\right(4){)}^{2}=4$ | ||
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. |
Try It 10.49
Solve $2{m}^{2}+16m-8=0$ by completing the square.
Try It 10.50
Solve $4{n}^{2}-24n-56=8$ by completing the square.
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 10.26
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. $\left(\frac{1}{2}\right(-\frac{3}{2}){)}^{2}=\frac{9}{16}$ | ||
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. |
Try It 10.51
Solve $3{r}^{2}-2r=21$ by completing the square.
Try It 10.52
Solve $4{t}^{2}+2t=20$ by completing the square.
Example 10.27
Solve $3{x}^{2}+2x=4$ 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 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 as a binomial square. | ||
Use the Square Root Property. | ||
Simplify the radical. | ||
Solve for x. | ||
Rewrite to show 2 solutions. | ||
Check. We leave the check for you. |
Try It 10.53
Solve $4{x}^{2}+3x=12$ by completing the square.
Try It 10.54
Solve $5{y}^{2}+3y=10$ by completing the square.
Media
Access these online resources for additional instruction and practice with solving quadratic equations by completing the square:
Practice Makes Perfect
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.
${b}^{2}+12b$
${n}^{2}+16n$
${n}^{2}-16n$
${q}^{2}-6q$
${y}^{2}+11y$
${q}^{2}+\frac{3}{4}q$
Solve Quadratic Equations of the Form ${x}^{2}+bx+c=0$ by Completing the Square
In the following exercises, solve by completing the square.
${w}^{2}+8w=65$
${z}^{2}+12z=\mathrm{-11}$
${d}^{2}-8d=9$
${y}^{2}-2y=8$
${n}^{2}-2n=\mathrm{-3}$
${t}^{2}-14t=\mathrm{-50}$
${b}^{2}+6b=41$
${z}^{2}+2z-5=2$
${w}^{2}=5w-1$
$\left(y+9\right)\left(y+7\right)=79$
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.
$2{n}^{2}+4n-26=0$
$3{d}^{2}-4d=15$
$3{q}^{2}-5q=9$
Everyday Math
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?
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 $\left(s+3\right)\left(s+9\right)=1080$ for $s$, the length of a side of the pool.
Writing Exercises
Solve the equation ${x}^{2}+10x=\mathrm{-25}$ ⓐ by using the Square Root Property and ⓑ by completing the square. ⓒ Which method do you prefer? Why?
Solve the equation ${y}^{2}+8y=48$ by completing the square and explain all your steps.
Self Check
ⓐ 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?