9.2 Solve Quadratic Equations by Completing the Square

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, which is important for our work on conics later.

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.

$\begin{matrix} {(y - 7)}^{2} & = & 12 \\ y - 7 & = & \pm \sqrt{12} \\ y - 7 & = & \pm 2 \sqrt{3} \\ y & = & 7 \pm 2 \sqrt{3} \end{matrix}$

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.

$\begin{matrix} x^{2} - 10 x + 25 & = & 18 \\ {(x - 5)}^{2} & = & 18 \end{matrix}$

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.

$\begin{matrix} {(x + 9)}^{2} & {(y - 7)}^{2} \\ (x + 9) (x + 9) & (y - 7) (y - 7) \\ x^{2} + 9 x + 9 x + 81 & y^{2} - 7 y - 7 y + 49 \\ x^{2} + 18 x + 81 & y^{2} - 14 y + 49 \end{matrix}$

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.

The perfect square expression a squared plus 2 a b plus b squared is shown above the expression x squared plus 6x plus an unknown to allow a comparison of the corresponding terms of the expressions.

We ultimately need to find the last term of this trinomial that will make it a perfect square trinomial. To do that we will need to find b. But first we start with determining a. Notice that the first term of x² + 6x is a square, x². This tells us that a = x.

The perfect square expression a squared plus 2 a b plus b squared is shown above the expression x squared plus 2 x b + b squared. Note that x has been substituted for a in the second equation and compare corresponding terms.

What number, b, when multiplied with 2x gives 6x? It would have to be 3, which is $\frac{1}{2} (6) .$ So b = 3.

The perfect square expression a squared plus 2 a b plus b squared is shown above the expression x squared plus 2 times 3 times x plus an unknown value to help compare terms.

Now to complete the perfect square trinomial, we will find the last term by squaring b, which is 3² = 9.

The perfect square expression a squared plus 2 a b plus b squared is shown above the expression x squared plus 6 x plus 9.

We can now factor.

The factored expression, the square of a plus b, is shown over the square of the expression x + 3.

So we found that adding 9 to x² + 6x ‘completes the square’, and we write it as (x + 3)².

Example 1

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

ⓐ $x squared minus 26 x$ ⓑ $y squared minus 9 y$ ⓒ $n^{2} + \frac{1}{2} n$

ⓐ


The coefficient of $x$ is −26.
$\begin{matrix} Find {(\frac{1}{2} b)}^{2} . \\ {(\frac{1}{2} \cdot (−26))}^{2} \\ {(13)}^{2} \\ 169 \end{matrix}$
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 $−9$ .
$\begin{matrix} Find {(\frac{1}{2} b)}^{2} . \\ {(\frac{1}{2} \cdot (−9))}^{2} \\ {(− \frac{9}{2})}^{2} \\ \frac{81}{4} \end{matrix}$
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 $\frac{1}{2} .$
$\begin{matrix} Find {(\frac{1}{2} b)}^{2} . \\ {(\frac{1}{2} \cdot \frac{1}{2})}^{2} \\ {(\frac{1}{4})}^{2} \\ \frac{1}{16} \end{matrix}$
Add $\frac{1}{16}$ to the binomial to complete the square.
Rewrite as a binomial square.

Solve Quadratic Equations of the Form x² + 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 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.

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.

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

We will start the next example by isolating the variable terms on the left side of the equation.

To solve the next equation, we must first collect all the variable terms on the left side of the equation. Then we proceed as we did in the previous examples.

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 since it says “If $a \cdot b = 0,$ then a = 0 or b = 0.” Instead, we multiply the factors and then put the equation into standard form to solve by completing the square.

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

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 8

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

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.

Now that we have seen that the coefficient of x² must be 1 for us to complete the square, we update our procedure for solving a quadratic equation by completing the square to include equations of the form ax² + bx + c = 0.

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.

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.

ⓐ $m squared minus 24 m$ ⓑ $x squared minus 11 x$ ⓒ $p^{2} - \frac{1}{3} p$

ⓐ $open parenthesis m minus 12 close parenthesis squared$ ⓑ ${(x - \frac{11}{2})}^{2}$
ⓒ ${(p - \frac{1}{6})}^{2}$

ⓐ $n squared minus 16 n$ ⓑ $y squared plus 15 y$ ⓒ $q^{2} + \frac{3}{4} q$

ⓐ $p squared minus 22 p$ ⓑ $y squared plus 5 y$ ⓒ $m^{2} + \frac{2}{5} m$

ⓐ $open parenthesis p minus 11 close parenthesis squared$ ⓑ ${(y + \frac{5}{2})}^{2}$
ⓒ ${(m + \frac{1}{5})}^{2}$

ⓐ $q squared minus 6 q$ ⓑ $x squared minus 7 x$ ⓒ $n^{2} - \frac{2}{3} n$

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

In the following exercises, solve by completing the square.

$u^{2} + 2 u = 3$

$u = −3, u = 1$

$z^{2} + 12 z = −11$

$x^{2} - 20 x = 21$

$x = −1, x = 21$

$y^{2} - 2 y = 8$

$m^{2} + 4 m = −44$

$m = −2 \pm 2 \sqrt{10} i$

$n^{2} - 2 n = −3$

$r^{2} + 6 r = −11$

$r = −3 \pm \sqrt{2} i$

$t^{2} - 14 t = −50$

$a^{2} - 10 a = −5$

$a = 5 \pm 2 \sqrt{5}$

$b^{2} + 6 b = 41$

$x^{2} + 5 x = 2$

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

$y^{2} - 3 y = 2$

$u^{2} - 14 u + 12 = −1$

$u = 1, u = 13$

$z^{2} + 2 z - 5 = 2$

$r^{2} - 4 r - 3 = 9$

$r = −2, r = 6$

$t^{2} - 10 t - 6 = 5$

$v^{2} = 9 v + 2$

$v = \frac{9}{2} \pm \frac{\sqrt{89}}{2}$

$w^{2} = 5 w - 1$

$x^{2} - 5 = 10 x$

$x = 5 \pm \sqrt{30}$

$y^{2} - 14 = 6 y$

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

$x = −7, x = 3$

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

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

$x = −5, x = −1$

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

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

In the following exercises, solve by completing the square.

$3 m^{2} + 30 m - 27 = 6$

$m = −11, m = 1$

$2 x^{2} - 14 x + 12 = 0$

$2 n^{2} + 4 n = 26$

$n = - 1 \pm \sqrt{14}$

$5 x^{2} + 20 x = 15$

$2 c^{2} + c = 6$

$c = −2, c = \frac{3}{2}$

$3 d^{2} - 4 d = 15$

$2 x^{2} + 7 x - 15 = 0$

$x = −5, x = \frac{3}{2}$

$3 x^{2} - 14 x + 8 = 0$

$2 p^{2} + 7 p = 14$

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

$3 q^{2} - 5 q = 9$

$5 x^{2} - 3 x = −10$

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

$7 x^{2} + 4 x = −3$

Writing Exercises

Solve the equation $x^{2} + 10 x = −25$

ⓐ by using the Square Root Property

ⓑ by Completing the Square

Answers will vary.

Solve the equation $y^{2} + 8 y = 48$ 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.

This table provides a checklist to evaluate mastery of the objectives of this section. Choose how would you respond to the statement “I can complete the square of a binomial expression.” “Confidently,” “with some help,” or “No, I don’t get it.” Choose how would you respond to the statement “I can solve quadratic equations of the form x squared plus b times x plus c equals 0 by completing the square.” “Confidently,” “with some help,” or “No, I don’t get it.” Choose how would you respond to the statement “I can solve quadratic equations of the form a times x squared plus b times x plus c equals 0 by completing the square.” “Confidently,” “with some help,” or “No, I don’t get it.”

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