1. Solve Quadratic Equations by Completing the Square
Lesson 10.6
Holt: 9-7

2. Warm Up Part I
Simplify. 1. 2. 19 3. 4.

3. Warm Up Part II
Solve each quadratic equation by factoring.
5. x2 + 8x + 16 = 0
6. x2 – 22x = 0
7. x2 – 12x + 36 = 0 –4 11 6

4. California
Standards
14.0 Students solve a quadratic equation by factoring or completing the square.
23.0 Students apply quadratic equations to physical problems, such as the motion of an object under the force of gravity.
Also covered:
2.0

5. In the previous lesson, you solved quadratic equations by isolating x2 and then using square roots. This method also works if the quadratic equation, when written in standard form, is a perfect square.

6. When a trinomial is a perfect square, there is a relationship between the coefficient of the x-term and the constant term.
Divide the coefficient of the x-term by 2. Then square the result to get the constant term.
(x + n)2
(x – n)2
= x2 + 2nx + n2
= x2 – 2nx + n2

7. An expression in the form x2 + bx is not a perfect square
An expression in the form x2 + bx is not a perfect square. However, you can use the relationship shown on the previous slide to add a term to
x2 + bx to form a trinomial that is a perfect square. This is called completing the square.

8. Additional Example 1: Completing the Square
Complete the square to form a perfect square trinomial.
A. x2 + 2x +
B. x2 – 6x +
x2 + 2x
x2 + –6x
Identify b. .
x2 + 2x + 1
x2 – 6x + 9

9. Check It Out! Example 1
Complete the square to form a perfect square trinomial.
a. x2 + 12x +
b. x2 – 5x +
x2 + 12x
x2 + –5x
Identify b. .
x2 – 5x +
x2 + 12x + 36

10. Check It Out! Example 1
Complete the square to form a perfect square trinomial.
c. 8x + x2 +
x2 + 8x
Identify b. .
x2 + 8x + 16

11. To solve a quadratic equation in the form x2 + bx = c, first complete the square of x2 + bx. Then you can solve using square roots.

12. Solving a Quadratic Equation by Completing the Square

13. Additional Example 2A: Solving x2 +bx = c by Completing the Square
Solve by completing the square.
x2 + 16x = –15
The equation is in the form x2 + bx = c.
Step 1 x2 + 16x = –15
Step 2 .
Step 3 x2 + 16x + 64 = –
Complete the square.
Step 4 (x + 8)2 = 49
Factor and simplify.
Step 5 x + 8 = ± 7
Take the square root of both sides.
Step 6 x + 8 = 7 or x + 8 = –7
x = –1 or x = –15
Write and solve two equations.

14. Additional Example 2A Continued
Solve by completing the square.
x2 + 16x = –15
Check
x2 + 16x = –15
(–1)2 + 16(–1) –15
1 – –15
– –15 
x2 + 16x = –15
(–15)2 + 16(–15) –15
225 – –15
– –15 

15. Additional Example 2B: Solving x2 +bx = c by Completing the square
Solve by completing the square.
x2 – 4x – 6 = 0
Write in the form x2 + bx = c.
Step 1 x2 + (–4x) = 6
Step 2 .
Step 3 x2 – 4x + 4 = 6 + 4
Complete the square.
Step 4 (x – 2)2 = 10
Factor and simplify.
Take the square root of both sides.
Step 5 x – 2 = ± √10
Step 6 x – 2 = √10 or x – 2 = –√10
Write and solve two equations.
x = 2 √10

16. The expressions and can be written as one expression: , which is read as “1 plus or minus the square root of 3.”
Writing Math

17. The equation is in the form x2 + bx = c.
Check It Out! Example 2a
Solve by completing the square.
x2 + 10x = –9
The equation is in the form x2 + bx = c.
Step 1 x2 + 10x = –9
Step 2 .
Step 3 x2 + 10x + 25 = –9 + 25
Complete the square.
Factor and simplify.
Step 4 (x + 5)2 = 16
Take the square root of both sides.
Step 5 x + 5 = ± 4
Step 6 x + 5 = 4 or x + 5 = –4
x = –1 or x = –9
Write and solve two equations.

18. Check It Out! Example 2a Continued
Solve by completing the square. Check your answer.
x2 + 10x = –9
Check
x2 + 10x = –9
(–1)2 + 10(–1) –9
1 – –9
– –9 
x2 + 10x = –9
(–9)2 + 10(–9) –9
81 – –9
–9 –9 

19. Take the square root of both sides. Step 5 t – 4 = ± √21
Check It Out! Example 2b
Solve by completing the square.
t2 – 8t – 5 = 0
Write in the form
x2 + bx = c.
Step 1 t2 + (–8t) = 5
Step 2 .
Step 3 t2 – 8t + 16 =
Complete the square.
Factor and simplify.
Step 4 (t – 4)2 = 21
Take the square root of both sides.
Step 5 t – 4 = ± √21
Step 6 t = 4 + √21 or t = 4 – √21
Write and solve two equations.

20. Additional Example 3A: Solving ax2 + bx = c by Completing the Square
Solve by completing the square.
–3x2 + 12x – 15 = 0
Divide both sides of the equation by – 3 so that a = 1.
Step 1
x2 – 4x + 5 = 0
x2 – 4x = –5
Write in the form x2 + bx = c.
x2 + (–4x) = –5
Step 2 .
Step 3
x2 – 4x + 4 = –5 + 4
Complete the square by adding 4 to both sides.

21. Additional Example 3A Continued
Solve by completing the square.
–3x2 + 12x – 15 = 0
Step 4
(x – 2)2 = –1
Factor and simplify. ø
There is no real number whose square is negative, so there are no real solutions.

22. Additional Example 3B: Solving ax2 + bx = c by Completing the Square
Solve by completing the square.
5x2 + 19x = 4
Step 1
Divide both sides of the equation by 5 so that a = 1.
Write in the form x2 + bx = c. .
Step 2

23. Additional Example 3B Continued
Solve by completing the square.
Complete the square by adding to both sides.
Step 3
Rewrite using like denominators.
Step 4
Factor and simplify.

24. Additional Example 3B Continued
Solve by completing the square.
Take the square root of both sides.
Step 5
Step 6
Write and solve two equations.

25. Check It Out! Example 3a
Solve by completing the square. Check your answer.
3x2 – 5x – 2 = 0
Divide both sides of the equation by 3 so that a = 1.
Step 1
Write in the form x2 + bx = c.

26. Check It Out! Example 3a Continued
Solve by completing the square. Check your answer. .
Step 2
Step 3
Complete the square by adding to both sides.
Step 4
Factor and simplify.

27. Check It Out! Example 3a Continued
Solve by completing the square. Check your answer.
Step 5
Take the square root of both sides.
Step 6
Write and solve two equations.

28. Check It Out! Example 3a Continued
Solve by completing the square. Check your answer.
3x2 – 5x – 2 = 0
Check
3(2)2 – 5(2) – 2 0
12 – 10 – 2 0 
3x2 – 5x – 2 = 0 
3x2 – 5x – 2 = 0
– – 2 0

29. Check It Out! Example 3b
Solve by completing the square.
4t2 – 4t + 9 = 0
Step 1
Divide by 4 to make a = 1.
Write in the form x2 + bx = c.

30. Check It Out! Example 3b Continued
Solve by completing the square.
4t2 – 4t + 9 = 0 .
Step 2
Step 3
Complete the square.
Step 4
Factor and simplify.
There is no real number whose square is negative, so there are no real solutions.

31. Understand the Problem
Additional Example 4: Problem-Solving Application
A rectangular room has an area of 195 square feet. Its width is 2 feet shorter than its length. Find the dimensions of the room. Round to the nearest hundredth of a foot, if necessary. 1
Understand the Problem
The answer will be the length and width of the room.
List the important information:
The room area is 195 square feet.
The width is 2 feet less than the length.

32. Additional Example 4 Continued2
Make a Plan
Set the formula for the area of a rectangle equal to 195, the area of the room.
Solve the equation.

33. Additional Example 4 Continued
Solve 3
Let x be the width. Then x + 2 is the length.
Use the formula for area of a rectangle.
l • w = A
length
times
width =
area of room
x + 2 • x =
195

34. Additional Example 4 Continued
Step 1 x2 + 2x = 195
Simplify. .
Step 2
Complete the square by adding 1 to both sides.
Step 3 x2 + 2x + 1 =
Factor the perfect-square trinomial.
Step 4 (x + 1)2 = 196
Take the square root of both sides.
Step 5 x + 1 = ±14

35. Additional Example 4 Continued
Step 6 x + 1 = 14 or x + 1 = –14
Write and solve two equations.
x = 13 or x = –15
Negative numbers are not reasonable for length, so x = 13 is the only solution that makes sense.
The width is 13 feet, and the length is , or 15 feet. 4
Look Back
The length of the room is 2 feet greater than the width. Also 13(15) = 195.

36. Understand the Problem
Check It Out! Example 4
A rectangular room has an area of 400 ft2. The length is 8 ft longer than the width. Find the dimensions of the room. Round to the nearest tenth of a foot. 1
Understand the Problem
The answer will be the length and width of the room.
List the important information:
The room area is 400 square feet.
The length is 8 feet more than the width.

37. Check It Out! Example 4 Continued2
Make a Plan
Set the formula for the area of a rectangle equal to 400, the area of the room.
Solve the equation.

38. Check It Out! Example 4 Continued
Solve 3
Let x be the width.
Then x + 8 is the length.
Use the formula for area of a rectangle.
l • w = A
length
times
width =
area of room
X + 8 • x =
400

39. Check It Out! Example 4 Continued
Step 1 x2 + 8x = 400
Simplify.
Step 2 .
Step 3 x2 + 8x + 16 =
Complete the square by adding 16 to both sides.
Step 4 (x + 4)2 = 416
Factor the perfect-square trinomial.
Take the square root of both sides.
Step 5 x + 4  ±20.4
Step 6 x + 4  20.4 or x + 4  –20.4
Write and solve two equations.
x  16.4 or x  –24.4

40. Check It Out! Example 4 Continued
Negative numbers are not reasonable for length, so x  16.4 is the only solution that makes sense.
The width is approximately 16.4 feet, and the length is , or approximately 24.4 feet. 4
Look Back
The length of the room is 8 feet longer than the width. Also 16.4(24.4) = , which is approximately 400 ft2.

41. Lesson Quiz: Part I
Complete the square to form a perfect square trinomial.
1. x2 + 11x +
2. x2 – 18x +
Solve by completing the square.
3. x2 – 2x – 1 = 0
4. 3x2 + 6x = 144
5. 4x x = 23 81
6, –8

42. Lesson Quiz: Part II
6. Dymond is painting a rectangular banner for a football game. She has enough paint to cover 120 ft2. She wants the length of the banner to be 7 ft longer than the width. What dimensions should Dymond use for the banner?
8 feet by 15 feet