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2.2.3: Solving Quadratic Equations by Completing the Square

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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 x2+bx+c=0 by completing the square
  • Solve quadratic equations of the form ax2+bx+c=0 by completing the square

Be Prepared

Before you get started, take this readiness quiz.

  1. Expand (x+9)2.
  2. Factor y214y+49.
  3. Factor 5n2+40n+80.

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.

Exercise 2.2.3.1

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Answer

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Example 2.2.3.1

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Solution

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Exercise 2.2.3.1

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Answer

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Complete the Square of a Binomial Expression

In the last section, we were able to use the Square Root Property to solve the equation (y7)2=12 because the left side was a perfect square.

(y7)2=12y7=±12y7=±23y=7±23

We also solved an equation in which the left side was a perfect square trinomial, but we had to rewrite it the form (xk)2 in order to use the Square Root Property. 

x210x+25=18(x5)2=18x5=±32x=5±32

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.

(x+9)2=(x+9)(x+9)=x2+9x+9x+81=x2+18x+81(y7)2=(y7)(y7)=y27y7y+49=y214y+49

We restate the patterns here for reference.

Binomial Squares Pattern

If a and b are real numbers,

(a+b)2=a2+2ab+b2,

and

(ab)2=a22ab+b2.

  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.

x2+6x+a2+2ab+b2

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.

x2+2xb+b2a2+2ab+b2

What number, b, when multiplied with 2x gives 6x? It would have to be 3, which is 12(6). So b=3.

x2+23x+a2+2ab+b2

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

x2+6x+9a2+2ab+b2

We can now factor.

(x+3)2(a+b)2

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

Complete a Square of x2+bx

  1. Identify b, the coefficient of x.
  2. Find (12b)2, the number to complete the square.
  3. Add the(12b)2 to x2+bx.
  4. Factor the perfect square trinomial, writing it as a binomial squared.

Example 2.2.3.1

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

a. x226x

b. y29y

c. n2+12n

Solution

a.

  x226xx2+bx
The coefficient of x is 26. b=26

Find (12b)2.

(12(26))2
=(13)2
=169
Add 169 to the binomial to complete the square.

x226x+169

Factor the perfect square trinomial, writing it as a binomial squared. =(x13)2

b.

  y29yx2+bx
The coefficient of y is 9. b=9

Find (12b)2.

 

(12(9))2
=(92)2
=814
Add 814 to the binomial to complete the square.

y29y+814

Factor the perfect square trinomial, writing it as a binomial squared.

=(y92)2

c.

  n2+12nx2+bx
The coefficient of n is 12. b=12

Find (12b)2.

 

(1212)2
=(14)2
=116
Add 116 to the binomial to complete the square. n2+12n+116
Rewrite as a binomial square. =(n+14)2

Try It 2.2.3.2

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

a. a220a

b. m25m

c. p2+14p

Answer

a. (a10)2

b. (m52)2

c. (p+18)2

Try It 2.2.3.3

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

a. b24b

b. n2+13n

c. q223q

Answer

a. (b2)2

b. (n+132)2

c. (q13)2

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.

  x2+6x=40
  x2+6x+=40+
Add 9 to both sides to complete the square. x2+6x+9=40+9
Rewrite it as a binomial square. (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.

Example 2.2.3.4

Solve by completing the square: x2+8x=48.

Solution
    x2+8x=48
Isolate the variable terms on one side and the constant terms on the other. This equation has all the variables on the left. x2+bxcx2+8x=48
Find (12b)2, the number to complete the square. Add it to both sides of the equation.

b=8

Take half of 8 and square it.

42=16

Add 16 to BOTH sides of the equation.

x2+8x+(128)2=48

x2+8x+16=48+16

Factor the perfect square trinomial as a binomial square.

x2+8x+16=(x+4)2

Add the terms on the right.

(x+4)2=64
Use the Square Root Property.   x+4=±64
Simplify the radical.  

x+4=±8

 

Solve the two resulting equations.   x+4=8x=4x+4=8x=12
Check the solutions. Put each answer in the original equation to check. Substitute x=4 and x=12.

x2+8x=48x=4:(4)2+8(4)?=4816+32?=4848?=48True

 

x2+8x=48x=12:(12)2+8(12)?=4814496?=4848?=48True

 

    The solutions are x=4 or x=12.

Try It 2.2.3.5

Solve by completing the square: x2+4x=5.

Answer

x=5 or x=1 

Try It 2.2.3.6

Solve by completing the square: y210y=9.

Answer

y=1 or y=9

The steps to solve a quadratic equation by completing the square are listed here.

Solve a Quadratic Equation of the Form x2+bx+c=0 by Completing the Square

  1. Isolate the variable terms on one side and the constant terms on the other.
  2. Find (12b)2, the number needed to complete the square. Add it to both sides of the equation.
  3. Factor the perfect square trinomial, writing it as a binomial squared on the left and simplify by adding the terms on the right.
  4. Use the Square Root Property.
  5. Simplify the radical and then solve the two resulting equations.
  6. Check the solutions.

When we solve an equation by completing the square, the answers will not always be integers.

Example 2.2.3.7

Solve by completing the square: x2+4x=21.

Solution
  x2+4x=21
The variable terms are on the left side. x2+bxcx2+4x=21

Take half of 4 and square it.

(12(4))2=4

 x2+4x+(124)2=21

Add 4 to both sides.

x2+4x+4=21+4

Factor the perfect square trinomial, writing it as a binomial squared. (x+2)2=17
Use the Square Root Property. x+2=±17
Simplifying using complex numbers. x+2=±17i
Subtract 2 from each side.

x=2±17i

Rewrite to show two solutions.

x=2+17i or x=217i

Check. We leave the check to you.
  The solutions are x=2+17i or x=217i.

Try It 2.2.3.8

Solve by completing the square: y210y=35.

Answer

y=5+10i or y=515i

Try It 2.2.3.9

Solve by completing the square: z2+8z=19.

Answer

z=4+3i or z=43i

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

Example 2.2.3.10

Solve by completing the square: y218y=6.

Solution
  y218y=6
The variable terms are on the left side.

x2+bxcy218y=6

Take half of 18 and square it.

(12(18))2=81

 y218y+(12(18))2=6

Add 81 to both sides.

y218y+81=6+81

Factor the perfect square trinomial, writing it as a binomial squared.

(y9)2=75

Use the Square Root Property.

y9=±75

Simplify the radical.

y9=±53

Solve for y.

y=9±53

Rewrite to show two solutions. y=9+53 or y=953 

Check.

.

Another way to check this would be to use a calculator. Evaluate y218y for both solutions. The answer should be 6.

Try It 2.2.3.11

Solve by completing the square: x216x=16.

Answer

x=8+43 or x=843

Try It 2.2.3.12

Solve by completing the square: y2+8y=11.

Answer

y=4+33 or y=433

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

Example 2.2.3.13

Solve by completing the square: x2+10x+4=15.

Solution
  x2+10x+4=15
Isolate the variable terms on the left side. Subtract 4 to get the constant terms on the right side. x2+10x=11

Take half of 10 and square it.

(12(10))2=25

 x210x+(12(10))2=11

Add 25 to both sides. x2+10x+25=11+25
Factor the perfect square trinomial, writing it as a binomial squared. (x+5)2=36
Use the Square Root Property. x+5=±36
Simplify the radical. x+5=±6
Solve for x. x=5±6
Rewrite to show two solutions. x=5+6 or x=56
Solve the equations. x=1 or x=11

Check.

.
  The solutions are x=1 or x=11.

Try It 2.2.3.14

Solve by completing the square: a2+4a+9=30.

Answer

a=7 or a=3

Try It 2.2.3.15

Solve by completing the square: b2+8b4=16.

Answer

b=10 or b=2

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 2.2.3.16

Solve by completing the square: n2=3n+11.

Answer
  n2=3n+11
Subtract 3n to get the variable terms on the left side. n23n=11
Take half of 3 and square it. (12(3))2=94
 

 n218y+(12(18))2=6

Add 94 to both sides. n23n+94=11+94
Factor the perfect square trinomial, writing it as a binomial squared.

(n32)2=444+94

Add the fractions on the right side. (n32)2=534
Use the Square Root Property. n32=±534
Simplify the radical. n32=±532 
Solve for n.

n=32±532 

Rewrite to show two solutions. n=32+532 or n=32532  

Check.

 

We leave the check for you!

Try It 2.2.3.17

Solve by completing the square: p2=5p+9.

Answer

p=52+612 or p=52612

Try It 2.2.3.18

Solve by completing the square: q2=7q3.

Answer

q=72+372 or q=72372

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 ab=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 2.2.3.19

Solve by completing the square: (x3)(x+5)=9.

Solution
  (x3)(x+5)=9
We multiple the binomials on the left. x2+2x15=9
Add 15 to isolate the constant terms on the right. x2+2x=24
   x2+2x+(12(2))2=24
Take half of 2 and square it. (12(2))2=1
Add 1 to both sides. x2+2x+1=24+1
Factor the perfect square trinomial, writing it as a binomial squared. (x+1)2=25
Use the Square Root Property. x+1=±25
Solve for x. x=1±5
Rewrite to show two solutions. x=1+5 or x=16
Simplify. x=4 or x=6

Check.

We leave the check for you!

Try It 2.2.3.20

Solve by completing the square: (c2)(c+8)=11.

Answer

c=9 or c=3

Try It 2.2.3.21

Solve by completing the square: (d7)(d+3)=56.

Answer

d=11 or =7

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 2.2.3.22

Solve by completing the square: 3x212x15=0.

Solution

To complete the square, we need the coefficient of x2 to be one. If we factor out the coefficient of x2 as a common factor, we can continue with solving the equation by completing the square.

  3x212x15=0
Factor out the greatest common factor. 3(x24x5)=0
Divide both sides by 3 to isolate the trinomial with coefficient 1. 3(x24x5)3=03
Simplify. x24x5=0
Add 5 to get the constant terms on the right side. x24x=5
Take half of 4 and square it. (12(4))2=4
   x24x+(12(4))2=5
Add 4 to both sides. x24x+4=5+4
Factor the perfect square trinomial, writing it as a binomial squared. (x2)2=9
Use the Square Root Property. x2=±9\
Solve for x. x2=±3
Rewrite to show two solutions.

x=2+3 or x=23

Simplify. x=5 or x=1

Check.

.

Try It 2.2.3.23

Solve by completing the square: 2m2+16m+14=0.

Answer

m=7 or m=1

Try It 2.2.3.24

Solve by completing the square: 4n224n56=8.

Answer

n=2 or n=8

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.

Example 2.2.3.25

Solve by completing the square: 2x23x=20.

Solution

To complete the square we need the coefficient of x2 to be one. We will divide both sides of the equation by the coefficient of x2. Then we can continue with solving the equation by completing the square.

  2x23x=20
Divide both sides by 2 to get the coefficient of x2 to be 1. 2x23x2=202
Simplify. x232x=10
Take half of 32 and square it. (12(32))2=916
  x232x+\( x24x+(12(4))2=5.
Add 916 to both sides. x232x+916=10+916
Factor the perfect square trinomial, writing it as a binomial squared. (x34)2=16916+916
Add the fractions on the right side. (x34)2=16016+916
Use the Square Root Property. x34=±16916
Simplify the radical. x34=±134
Solve for x. x=34±134
Rewrite to show two solutions. x=34+134 or x=34±134
Simplify. x=4 or x=52
Check. We leave the check for you!

Try It 2.2.3.26

Solve by completing the square: 3r22r=21.

Answer

r=73 or r=3

Try It 2.2.3.27

Solve by completing the square: 4t2+2t=20.

Answer

t=52 or t=2

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.

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

  1. Divide by aa 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 (12b)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.

Example 2.2.3.28

Solve by completing the square: 3x2+2x=4.

Solution

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

  3x2+2x=4
Divide both sides by 3 to make the coefficient of x2 equal 1. .
Simplify. .
Take half of 23 and square it.  
(1223)2=19  
  .
Add 19 to both sides. .
Factor the perfect square trinomial, writing it as a binomial squared. .
Use the Square Root Property. .
Simplify the radical. .
Solve for x. .
Rewrite to show two solutions. .
Check. We leave the check for you!

Try It 2.2.3.29

Solve by completing the square: 4x2+3x=2.

Answer

x=38+418 or x=38418

Try It 2.2.3.30

Solve by completing the square: 3y210y=5.

Answer

y=53+103 or y=53103

 

Exit Problem 2.2.3.31

Solve x2+x=5 by completing the square and applying the Square Root Property.

Key Concepts

  • Binomial Squares Pattern
    If a and b are real numbers,
Quantity a plus b squared equals a squared plus 2 a b plus b2 where the binomial squared equals the first term squared plus 2 times the product of terms plus the second term squared. Quantity a minus b squared equals a squared minus 2 a b plus b2 where the binomial squared equals the first term squared minus 2 times the product of terms plus the second term squared.
  • How to Complete a Square
    1. Identify b, the coefficient of x.
    2. Find (12b)2, the number to complete the square.
    3. Add the (12b)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 (12b)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.

This page titled 2.2.3: Solving Quadratic Equations by Completing the Square is shared under a CC BY license and was authored, remixed, and/or curated by Holly Carley and Ariane Masuda.

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