Elementary Algebra 2e · free preview

10.2 Solve Quadratic Equations by Completing the Square

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 $a{x}^{2}+bx+c=0$ by completing the square

1. Learning objectives

Learning objectives

Learning objectives

  • Complete the square of a binomial expression
  • Solve quadratic equations of the form x2+bx+c=0{x}^{2}+bx+c=0 by completing the square
  • Solve quadratic equations of the form ax2+bx+c=0a{x}^{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.’

2. 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{(y-7)}^{2}=12 because the left side was a perfect square.

(y−7)2=12y−7=± 12y−7=± 23y=7±23\begin{array}{rll} {(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{(x-k)}^{2} in order to use the square root property.

x2−10x+25=18(x−5)2=18\begin{array}{rll} {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.

(x+9)2(x+9)(x+9)x2+9x+9x+81x2+18x+81 (y−7)2(y−7) (y−7)y2−7y−7y+49y2−14y+49\begin{array}{lccl} \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}

If a,ba,b are real numbers,

(a+b)2=a2+2ab+b2{(a+b)}^{2}={a}^{2}+2ab+{b}^{2}
(a−b)2=a2−2ab+b2{(a-b)}^{2}={a}^{2}-2ab+{b}^{2}

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

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

a2+2ab+b2=(a+b)2{a}^{2}+2ab+{b}^{2}={(a+b)}^{2}

Notice that the first term of x2+6x{x}^{2}+6x is a square, x2{x}^{2}.

We now know a=xa=x.

What number can we add to x2+6x{x}^{2}+6x to make a perfect square trinomial?

The middle term of the Binomial Squares Pattern, 2ab2ab, is twice the product of the two terms of the binomial. This means twice the product of xx and some number is 6x6x. So, two times some number must be six. The number we need is 12⋅6=3.\frac{1}{2}\cdot{}6=3. The second term in the binomial, b,b, must be 3.

We now know b=3b=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 x2+6x{x}^{2}+6x ‘completes the square,’ and we write it as (x+3)2{(x+3)}^{2}.

Complete a square.

To complete the square of x2+bx{x}^{2}+bx:

  1. Step 1. Identify bb, the coefficient of xx.
  2. Step 2. Find (12b)2{(\frac{1}{2}b)}^{2}, the number to complete the square.
  3. Step 3. Add the (12b)2{(\frac{1}{2}b)}^{2} to x2+bx{x}^{2}+bx.

3. Example 10.14

Example 10.14

Example 10.14

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

x2+14x{x}^{2}+14x

Solution

4. Example 10.15

Example 10.15

Example 10.15

Complete the square to make a perfect square trinomial. Then, write the result as a binomial squared. m2−26m{m}^{2}-26m

Solution

5. Example 10.16

Example 10.16

Example 10.16

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

u2−9u{u}^{2}-9u

Solution

6. Example 10.17

Example 10.17

Example 10.17

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

p2+12p{p}^{2}+\frac{1}{2}p

Solution

7. Solve Quadratic Equations of the Form *x*${}^{2}$ + *bx* + *c* = 0 by completing the square

Solve Quadratic Equations of the Form x2{}^{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 x2+6x=40{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{(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.

8. Example 10.18

Example 10.18

Example 10.18

How To Solve a Quadratic Equation of the Form x2+bx+c=0{x}^{2}+bx+c=0 by Completing the Square

Solve x2+8x=48{x}^{2}+8x=48 by completing the square.

Solve a quadratic equation of the form x2+bx+c=0{x}^{2}+bx+c=0 by completing the square.

  1. Step 1. Isolate the variable terms on one side and the constant terms on the other.
  2. Step 2. Find (12⋅b)2{(\frac{1}{2}\cdot{}b)}^{2}, the number to complete the square. Add it to both sides of the equation.
  3. Step 3. Factor the perfect square trinomial as a binomial square.
  4. Step 4. Use the Square Root Property.
  5. Step 5. Simplify the radical and then solve the two resulting equations.
  6. Step 6. Check the solutions.

9. Example 10.19

Example 10.19

Example 10.19

Solve y2−6y=16{y}^{2}-6y=16 by completing the square.

Solution

10. Example 10.20

Example 10.20

Example 10.20

Solve x2+4x=−21{x}^{2}+4x=-21 by completing the square.

Solution

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

11. Example 10.21

Example 10.21

Example 10.21

Solve p2−18p=−6{p}^{2}-18p=-6 by completing the square.

Solution

Another way to check this would be to use a calculator. Evaluate p2−18p{p}^{2}-18p for both of the solutions. The answer should be −6-6.

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

12. Example 10.22

Example 10.22

Example 10.22

Solve x2+10x+4=15{x}^{2}+10x+4=15 by completing the square.

Solution

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.

13. Example 10.23

Example 10.23

Example 10.23

Solve n2=3n+11{n}^{2}=3n+11 by completing the square.

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. Instead, we multiply the factors and then put the equation into the standard form to solve by completing the square.

14. Example 10.24

Example 10.24

Example 10.24

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

Solution

15. Solve Quadratic Equations of the form *ax*${}^{2}$ + *bx* + *c* = 0 by completing the square

Solve Quadratic Equations of the form ax2{}^{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 x2+bx+c{x}^{2}+bx+c. If the x2{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.

16. Example 10.25

Example 10.25

Example 10.25

Solve 3x2−12x−15=03{x}^{2}-12x-15=0 by completing the square.

Solution

To complete the square, we need the coefficient of x2{x}^{2} to be one. If we factor out the coefficient of x2{x}^{2} as a common factor, we can continue with solving the equation 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.

17. Example 10.26

Example 10.26

Example 10.26

Solve 2x2−3x=202{x}^{2}-3x=20 by completing the square.

Solution

Again, our first step will be to make the coefficient of x2{x}^{2} be one. By dividing both sides of the equation by the coefficient of x2{x}^{2}, we can then continue with solving the equation by completing the square.

18. Example 10.27

Example 10.27

Example 10.27

Solve 3x2+2x=43{x}^{2}+2x=4 by completing the square.

Solution

Again, our first step will be to make the coefficient of x2{x}^{2} be one. By dividing both sides of the equation by the coefficient of x2{x}^{2}, we can then continue with solving the equation by completing the square.

19. Section 10.2 Exercises

Section 10.2 Exercises

Section 10.2 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 x2+bx+c=0{x}^{2}+bx+c=0 by Completing the Square

In the following exercises, solve by completing the square.

Solve Quadratic Equations of the Form ax2+bx+c=0a{x}^{2}+bx+c=0 by Completing the Square

In the following exercises, solve by completing the square.

Everyday Math

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?

20. Practice 1

  1. Complete the square to make a perfect square trinomial. Write the result as a binomial square.

y2+12y{y}^{2}+12y

  1. Complete the square to make a perfect square trinomial. Write the result as a binomial square.

z2+8z{z}^{2}+8z

  1. Complete the square to make a perfect square trinomial. Write the result as a binomial square.

a2−20a{a}^{2}-20a

  1. Complete the square to make a perfect square trinomial. Write the result as a binomial square.

b2−4b{b}^{2}-4b

21. Practice 2

  1. Complete the square to make a perfect square trinomial. Write the result as a binomial square.

m2−5m{m}^{2}-5m

  1. Complete the square to make a perfect square trinomial. Write the result as a binomial square.

n2+13n{n}^{2}+13n

  1. Complete the square to make a perfect square trinomial. Write the result as a binomial square.

p2+14p{p}^{2}+\frac{1}{4}p

  1. Complete the square to make a perfect square trinomial. Write the result as a binomial square.

q2−23q{q}^{2}-\frac{2}{3}q

22. Practice 3

  1. Solve c2+4c=5{c}^{2}+4c=5 by completing the square.
  1. Solve d2+10d=−9{d}^{2}+10d=-9 by completing the square.
  1. Solve r2−4r=12{r}^{2}-4r=12 by completing the square.
  1. Solve t2−10t=11{t}^{2}-10t=11 by completing the square.

23. Practice 4

  1. Solve y2−10y=−35{y}^{2}-10y=-35 by completing the square.
  1. Solve z2+8z=−19{z}^{2}+8z=-19 by completing the square.
  1. Solve x2−16x=−16{x}^{2}-16x=-16 by completing the square.
  1. Solve y2+8y=11{y}^{2}+8y=11 by completing the square.

24. Practice 5

  1. Solve a2+4a+9=30{a}^{2}+4a+9=30 by completing the square.
  1. Solve b2+8b−4=16{b}^{2}+8b-4=16 by completing the square.
  1. Solve p2=5p+9{p}^{2}=5p+9 by completing the square.
  1. Solve q2=7q−3{q}^{2}=7q-3 by completing the square.

25. Practice 6

  1. Solve (c−2)(c+8)=7(c-2)(c+8)=7 by completing the square.
  1. Solve (d−7)(d+3)=56(d-7)(d+3)=56 by completing the square.
  1. Solve 2m2+16m−8=02{m}^{2}+16m-8=0 by completing the square.
  1. Solve 4n2−24n−56=84{n}^{2}-24n-56=8 by completing the square.

26. Practice 7

  1. Solve 3r2−2r=213{r}^{2}-2r=21 by completing the square.
  1. Solve 4t2+2t=204{t}^{2}+2t=20 by completing the square.
  1. Solve 4x2+3x=124{x}^{2}+3x=12 by completing the square.
  1. Solve 5y2+3y=105{y}^{2}+3y=10 by completing the square.

27. Practice 8

  1. a2+10a{a}^{2}+10a
  1. m2+18m{m}^{2}+18m
  1. m2−24m{m}^{2}-24m
  1. p2−22p{p}^{2}-22p

28. Practice 9

  1. x2−9x{x}^{2}-9x
  1. p2−13p{p}^{2}-\frac{1}{3}p
  1. v2+6v=40{v}^{2}+6v=40
  1. u2+2u=3{u}^{2}+2u=3

29. Practice 10

  1. c2−12c=13{c}^{2}-12c=13
  1. x2−20x=21{x}^{2}-20x=21
  1. m2+4m=−44{m}^{2}+4m=-44
  1. r2+6r=−11{r}^{2}+6r=-11

30. Practice 11

  1. a2−10a=−5{a}^{2}-10a=-5
  1. u2−14u+12=−1{u}^{2}-14u+12=-1
  1. v2=9v+2{v}^{2}=9v+2
  1. (x+6)(x−2)=9(x+6)(x-2)=9

31. Practice 12

  1. 3m2+30m−27=63{m}^{2}+30m-27=6
  1. 2c2+c=62{c}^{2}+c=6
  1. 2p2+7p=142{p}^{2}+7p=14
  1. 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 p2+4p=320{p}^{2}+4p=320 for pp, the length of one side of the playground. What is the length of the other side?

32. Practice 13

  1. Solve the equation x2+10x=−25{x}^{2}+10x=-25 ⓐ by using the Square Root Property and ⓑ by completing the square. ⓒ Which method do you prefer? Why?

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