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Section 9.3 Complete the Square

Objective: Solve quadratic equations by completing the square.

When solving quadratic equations in the past we have used factoring to solve for our variable. This is exactly what is done in the next example.

Example 9.3.1.

x2+5x+6=0 Factor (x+3)(x+2)=0 Set each factor equal to zero x+3=0 or x+2=0 Solve each equation 3 32 2 x=3 or x=2 Our Solution 

However, the problem with factoring is all equations cannot be factored. Consider the following equation: x22x7=0. The equation cannot be factored, however there are two solutions to this equation, 1+22 and 122. To find these two solutions we will use a method known as completing the square. When completing the square we will change the quadratic into a perfect square which can easily be solved with the square root property. The next example reviews the square root property.

Example 9.3.2.

(x+5)2=18 Square root of both sides (x+5)2=±18 Simplify each radical x+5=±32 Subtract 5 from both sides 5 5 x=5±32 Our Solution 

To complete the square, or make our problem into the form of the previous example, we will be searching for the third term in a trinomial. If a quadratic is of the form x2+bx+c, and a perfect square, the third term, c, can be easily found by the formula (12b)2. This is shown in the following examples, where we find the number that completes the square and then factor the perfect square.

Example 9.3.3.

x2+8x+cc=(12b)2 and our b=8(128)2=42=16 The third term to complete the square is 16x2+8x+16 Our equation as a perfect square, factor (x+4)2 Our Solution 

Example 9.3.4.

x27x+cc=(12b)2 and our b=7(127)2=(72)2=494 The third term to complete the square is 494x211x+494 Our equation as a perfect square, factor (x72)2 Our Solution 

Example 9.3.5.

x2+53x+cc=(12b)2 and our b=53(1253)2=(56)2=2536 The third term to complete the square is 2536x2+53x+2536 Our equation as a perfect square, factor (x+56)2 Our Solution 

The process in the previous examples, combined with the even root property, is used to solve quadratic equations by completing the square. The following five steps describe the process used to complete the square, along with an example to demonstrate each step.

Problem 3x2+18x6=0
1. Separate constant term from variables $+6+63x2+18x=6
2. Divide each term by a $33x2+183x=63x2+6x=2
3. Find value to complete the square: (12b)2 (126)2=32=9
4. Add to both sides of equation $x2+6x=2+9+9x2+6x+9=11
5. Factor (x+3)2=11
Solve by even root property (x+3)2=±11x+3=±113 3x=3±11

World View Note: The Chinese in 200 BC were the first known culture group to use a method similar to completing the square, but their method was only used to calculate positive roots.

The advantage of this method is it can be used to solve any quadratic equation. The following examples show how completing the square can give us rational solu- tions, irrational solutions, and even complex solutions.

Example 9.3.6.

2x2+20x+48=0 Separate constant term from varaibles 48 48 Subtract 242x2+20x=48 Divide by a or 22x22+20x2=482 x2+10x=24 Find number to complete the square: (12b)2(1210)2=52=25 Add 25 to both sides of the equation x2+10x=24 +25+25 x2+10x+25=1 Factor (x+5)2=1 Solve with even root property (x+5)2=±1 Simplify roots x+5=±1 Subtract 5 from both sides 5 5  x=5±1 Evaluate x=4 or 6 Our Solution 

Example 9.3.7.

x23x2=0 Separate constant from variables +2 +2 Add 2 to both sides x23x=2 a=1, find number to complete the square (12b)2(123)2=(32)2=94 Add 94 to both sides, 21(44)+94=84+94=174 Need common denominator (4) on right x23x+94=84+94=174 Factor (x32)2=174 Solve using the even root property (x32)2=±174 Simplify roots x32=±172 Add 32 to both sides, +32 +32 we already have a common denominator x=3±172 Our Solution 

Example 9.3.8.

3x2=2x7 Separate constant from variables 2x 2x Subtract 2x from both sides 3x22x=7 Divide each term by a=33x232x3=73 x223x=73 Find number to complete the square (12b)2(1223)2=(13)2=19 Add 19 to both sides, 73(33)+19=219+19=209 Need common denominator (9) on right x223x+13=209 Factor (x13)2=209 Solve using the even root property (x13)2=±209 Simplify roots x13=±2i53 Add 13 to both sides, +13 +13 we already have a common denominator x=1+2i±53 Our Solution 

As several of the examples have shown, when solving by completing the square we will often need to use fractions and be comfortable finding common denominators and adding fractions together. Once we get comfortable solving by completing the square and using the five steps, any quadratic equation can be easily solved.

Exercises Exercises - Complete the Square

Exercise Group.

Find the value that completes the square and then rewrite as a perfect square.

1.

x230x+ 

2.

a224a+ 

3.

m236m+ 

4.

x234x+ 

5.

x215x+ 

6.

r219r+ 

8.

p217p+ 

Exercise Group.

Solve each equation by completing the square.

9.

x216x+55=0

10.

n28n12=0

11.

v28v+45=0

13.

6x2+12x+63=0

14.

3x26x+47=0

15.

5k210k+48=0

16.

8a2+16a1=0

17.

x2+10x57=4

18.

p216p52=0

19.

n216n+67=4

20.

m28m3=6

21.

2x2+4x+38=6

22.

6r2+12r24=6

23.

8b2+16b37=5

24.

6n212n14=4

25.

x2=10x29

27.

n2=21+10n

28.

a256=10a

30.

5x2=26+10x

32.

5n2=10n+15

33.

p28p=55

35.

7n2n+7=7n+6n2

37.

13b2+15b+44=5+7b2+3b

38.

3r2+12r+49=6r2

39.

5x2+5x=315x

42.

b2+7b33=0

43.

7x26x+40=0

45.

k27k+50=3

46.

a25a+25=3

47.

5x2+8x40=8

48.

2p2p+56=8

50.

n2n=41

51.

8r2+10r=55

52.

3x211x=18

53.

5n28n+60=3n+6+4n2

54.

4b215b+56=3b2

55.

2x2+3x5=4x2

56.

10v215v=27+4v26v