9.2 Solve Quadratic Equations by Completing the Square

Learning objectives

Learning objectives

• 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 ${ax}^{2}+bx+c=0$ 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.

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.

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.

We restate the patterns here for reference.

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

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

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 x${}^{2}$ + 6*x* is a square, x${}^{2}$. This tells us that a = x.

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

Now to complete the perfect square trinomial, we will find the last term by squaring b, which is $3^{2}$ = 9.

We can now factor.

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

Example 9.11

Solution

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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${}^{2}$ + 6*x* = 40, and we want to complete the square on the left, we will add 9 to both sides of the equation.

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 9.12

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 9.13

Solution

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

Example 9.14

Solution

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

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

Example 9.15

Solution

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.

Example 9.16

Solution

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.

Example 9.17

Solution

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 coefficient of x${}^{2}$ is 1, so the left side of the equation is of the form x${}^{2}$ + bx + c. If the x${}^{2}$ 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 9.18

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.

To complete the square, the coefficient of the x${}^{2}$ 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 9.19

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 x${}^{2}$. Then we can continue with solving the equation by completing the square.

Now that we have seen that the coefficient of x${}^{2}$ 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 ax${}^{2}$ + bx + c = 0.

Example 9.20

Solution

Again, our first step will be to make the coefficient of x${}^{2}$ 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.

Section 9.2 Exercises

Section 9.2 Exercises

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.

Solve Quadratic Equations of the form x${}^{2}$ + bx + c = 0 by Completing the Square

In the following exercises, solve by completing the square.

Solve Quadratic Equations of the form ax${}^{2}$ + bx + c = 0 by Completing the Square

In the following exercises, solve by completing the square.

Writing Exercises

ⓐ 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?

1. Solve by completing the square: ${y}^{2}-10y=-9.$

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

3. Solve by completing the square: ${z}^{2}+8z=-19.$

1. Solve by completing the square: ${x}^{2}-16x=-16.$

2. Solve by completing the square: ${y}^{2}+8y=11.$

3. Solve by completing the square: ${a}^{2}+4a+9=30.$

1. Solve by completing the square: ${b}^{2}+8b-4=16.$

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

3. Solve by completing the square: ${q}^{2}=7q-3.$

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

2. Solve by completing the square: $(d-7)(d+3)=56.$

3. Solve by completing the square: $2{m}^{2}+16m+14=0.$

1. Solve by completing the square: $4{n}^{2}-24n-56=8.$

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

3. Solve by completing the square: $4{t}^{2}+2t=20.$

1. Solve by completing the square: $4{x}^{2}+3x=2.$

2. Solve by completing the square: $3{y}^{2}-10y=-5.$

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

1. ⓐ ${p}^{2}-22p$ ⓑ ${y}^{2}+5y$ ⓒ ${m}^{2}+\frac{2}{5}m$

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

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

1. ${m}^{2}+4m=-44$

2. ${r}^{2}+6r=-11$

3. ${a}^{2}-10a=-5$

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

2. ${u}^{2}-14u+12=-1$

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

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

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

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

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

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

3. $2{n}^{2}+4n=26$

1. $2{c}^{2}+c=6$

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

3. $2{p}^{2}+7p=14$

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

ⓐ ${a}^{2}-20a$ ⓑ ${m}^{2}-5m$ ⓒ ${p}^{2}+\frac{1}{4}p$

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

ⓐ ${b}^{2}-4b$ ⓑ ${n}^{2}+13n$ ⓒ ${q}^{2}-\frac{2}{3}q$

3. Solve by completing the square: ${x}^{2}+4x=5.$

1. $5{x}^{2}-3x=-10$

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

ⓐ by using the Square Root Property

ⓑ by Completing the Square

ⓒ Which method do you prefer? Why?