Using Completing the Square to Solve Equations

9.3.2 Using Completing the Square to Solve Equations

9.3.2 • Using Completing the Square to Solve Equations

Activity

Completing the square can be used to solve quadratic equations.

Example

Solve $x^{2} + 8 x - 2 = 0$ .

To complete the square to solve an equation, follow these steps:

Step 1 - Write the equation in the form $a x^{2} + b x = c$ .

$x^{2} + 8 x = - 2$

Step 2 - Complete the square by finding $(\frac{b}{2})^{2}$ .

$(\frac{b}{2})^{2} = (\frac{8}{2})^{2} = 16$

Step 3 - Add the value to both sides of the equation.

$x^{2} + 8 x + 16 = - 2 + 16$

Step 4 - Factor the perfect square trinomial and simplify the constants.

$(x + 4)^{2} = 14$

Step 5 - Take the square root of each side (remember the plus-minus sign with the constant).

$\sqrt{(x + 4)^{2}} = \sqrt{14}$

$x + 4 = \pm \sqrt{14}$

Step 6 - Write two separate equations and solve each.

$x + 4 = \sqrt{14}$ or $x + 4 = - \sqrt{14}$ $x = - 4 + \sqrt{14}$ or $x = - 4 - \sqrt{14}$

Solve each equation by completing the square. In part a, select the value of $(\frac{b}{2})^{2}$ that will complete the square. In part b, choose the two roots.

1. $(x - 3) (x + 1) = 5$

a. What is the correct value of $(\frac{b}{2})^{2}$ that will complete the square?

$(\frac{b}{2})^{2} = 1$

b. Select two roots that solve the quadratic equation.

$x = - 2$
$x = 1$
$x = 8$
$x = 4$
$x = 9$

$x = - 2$ , $x = 4$

The correct answers are: $x = - 2$ , $x = 4$ .

2. $x^{2} + \frac{1}{2} x = \frac{3}{16}$

a. What is the correct value of $(\frac{b}{2})^{2}$ that will complete the square?

Compare your answer:

$(\frac{b}{2})^{2} = \frac{1}{16}$

b. Select two roots that solve the quadratic equation.

$x = \frac{1}{2}$
$x = \frac{1}{4}$
$x = - \frac{3}{4}$
$x = - \frac{1}{4}$
$x = 1$

$x = \frac{1}{4}$ , $x = - \frac{3}{4}$

The correct answers are: $x = \frac{1}{4}$ , $x = - \frac{3}{4}$ .

3. $x^{2} + 3 x + \frac{8}{4} = 0$

a. What is the correct value of $(\frac{b}{2})^{2}$ that will complete the square?

Compare your answer:

$(\frac{b}{2})^{2} = \frac{9}{4}$

b. Select two roots that solve the quadratic equation.

$x = - 2$
$x = \frac{1}{16}$
$x = 9$
$x = \frac{1}{4}$
$x = - 1$

$x = - 2$ , $x = - 1$

The correct answers are: $x = - 2$ , $x = - 1$ .

4. $(7 - x) (3 - x) + 3 = 0$

a. What is the correct value of $(\frac{b}{2})^{2}$ that will complete the square?

$(\frac{b}{2})^{2} = 25$

b. Select two roots that solve the quadratic equation.

$x = - 2$
$x = 6$
$x = 21$
$x = 4$
$x = 9$

$x = 6$ , $x = 4$

The correct answers are: $x = 6$ , $x = 4$ .

5. $x^{2} + 1.6 x + 0.63 = 0$

a. What is the correct value of $(\frac{b}{2})^{2}$ that will complete the square?

$(\frac{b}{2})^{2} = 0.64$

b. Select two roots that solve the quadratic equation.

$x = - 0.7$
$x = 0.4$
$x = 0.27$
$x = 0.7$
$x = - 0.9$

$x = - 0.7$ , $x = - 0.9$

The correct answers are: $x = - 0.7$ , $x = - 0.9$ .

Video: Solve by Completing the Square

Watch the following video to learn more about solving to complete the square.

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Self Check

After solving using completing the square, what are the zeros of the quadratic equation

x^2+5x+\frac{9}{4}=0

$x= \frac12$
$x= -\frac{1}{2}$ , $x=-\frac{9}{2}$
$x=\frac{25}{4}$
$x=\frac{1}{4}$ , $x=-\frac{19}{4}$

Additional Resources

Completing the Square with Fractions

Completing the square can be a useful method for solving quadratic equations in cases in which it is not easy to rewrite an expression in factored form. For example, let's solve this equation:

$x^{2} + 5 x - \frac{75}{4} = 0$

First, we'll add $\frac{75}{4}$ to each side to make things easier on ourselves.

$x^{2} + 5 x - \frac{75}{4} + \frac{75}{4} = 0 + \frac{75}{4}$

$x^{2} + 5 x = \frac{75}{4}$

To complete the square, take $\frac{1}{2}$ of the coefficient of the linear term 5, which is $\frac{5}{2}$ , and square it, which is $\frac{25}{4}$ . Add this to each side:

$x^{2} + 5 x + \frac{25}{4} = \frac{75}{4} + \frac{25}{4}$

$x^{2} + 5 x + \frac{25}{4} = \frac{100}{4}$

Notice that $\frac{100}{4}$ is equal to 25 and rewrite it:

$x^{2} + 5 x + \frac{25}{4} = 25$

Since the left side is now a perfect square, let's rewrite it:

$(x + \frac{5}{2})^{2} = 25$

For this equation to be true, one of these equations must be true:

$x + \frac{5}{2} = 5$ or $x + \frac{5}{2} = - 5$

To finish up, we can subtract $\frac{5}{2}$ from each side of the equal sign in each equation.

$x = 5 - \frac{5}{2}$ or $x = - 5 - \frac{5}{2}$

$x = \frac{5}{2}$ or $x = - \frac{15}{2}$

$x = 2 \frac{1}{2}$ or $x = - 7 \frac{1}{2}$

It takes some practice to become proficient at completing the square, but it makes it possible to solve many more equations than you could by methods you learned previously.

Try it

Completing the Square with Fractions

Solve $x^{2} + 3 x - 4 = 0$ using completing the square, then find the zeros of the quadratic equation.

Here is how to find the zeros after using completing the square:

Step 1 - Add the constant to the other side of the equation.

$x^{2} + 3 x = 4$

Step 2 - Complete the square.

$(\frac{3}{2})^{2} = \frac{9}{4}$

Step 3 - Add the constant to each side.

$x^{2} + 3 x + \frac{9}{4} = \frac{16}{4} + \frac{9}{4}$

Step 4 - Simplify and factor the perfect square trinomial.

$(x + \frac{3}{2})^{2} = \frac{25}{4}$

Step 5 - Take the square root of both sides of the equation.

$x + \frac{3}{2} = \pm \frac{5}{2}$

Step 6 - Write the two equations and solve.

$\begin{matrix} x + 32 = 52 & and & x + 32 = - 52 \\ x = 1 & x = - 4 \end{matrix}$