10-5 Solve Quadratic Equations
by Completing the Square
Name
Date
Solve: 2x2 3x 1 0 Use completing the square.
3
1
3 2
1
c
b
Write in the form x2 x a
a
x2 2 x 2
3
3 2
3
Add 4
()
x2 2 x ( 4 ) 2 ( 4 )
(x 34 )
2
1
1
x44
3
1
3
Take the square root of both sides.
1
x 4 4 or x 4 4
1
x 2 or
to each side to complete the square.
3
9
Write x2 2x 16 as the square of a binomial.
16
3
2
x 1
Write and solve two equations.
Apply the Subtraction Property of Equality; simplify.
?
Check: 2x2 3x 1 0
?
1
1
2( 2 )2 3( 2 ) 1 0
?
1
3
2210
?
2x2 3x 1 0
?
2(1)2 3(1) 1 0
2310
0 0 True
0 0 True
Solve each equation by completing the square. Then check. Check students’ work.
Copyright © by William H. Sadlier, Inc. All rights reserved.
1. x2 2x 35 0
2. x2 4x 96 0
x2 2x 35
x2 2x 1 35 1
(x 1)2 36
x 1 66
x 1 6 or x 1 –6
x 5 or
x –7
x2 4x 96
4x 4 96 4
(x 2)2 100; x 2 10
x 2 10 or x 2 10
x 8 or
x 12
5. x2 11x 30 0
x2
x2 6x 2
2
x 6x 9 2 9
(x 3)2 7; x 3 7
x 3 7 or x 3 – 7
x 3 7 or x 3 7
( )
)
( )
8. x2 11x 102 0
9x 36
9 2
9 2
2
36 x 9x 2
2
15
9 2 225
9
x
;x 2
2
4
2
9
15
9 15
or
x x 2
2
2
2
x 12 or
x 3
()
x2
(
x2 13x 22
13 2
13 2
13x 22 2
2
13
9
13 2 81
x
;x
2
2
2
4
13
9
13 9
or
x
x
2
2
2
2
x 11 or
x2
( )
x2
(
( )
)
9. 2x2 5x 3 0
11x 102
11 2
11 2
2
102 x 11x 2
2
11 2 529
11
23
x
;x
2
4
2
2
11
23
11 23
or x x
2
2
2
2
x 17 or
x 6
( )
x2
6. x2 13x 22 0
11x 30
11 2
11 2
2
x 11x 30 2
2
11 2 1
11
1
x
;x
2
4
2
2
11
11 1
1
or
x
x
2
2
2
2
x 6 or
x5
(
7. x2 9x 36 0
x2
x2 4x 1
4x 4 1 4
(x 2)2 3; x 2 3
x 2 3 or x 2 3
x 2 3 or x 2 3
x2
4. x2 6x 2 0
()
( )
3. x2 4x 1 0
( )
)
Lesson 10-5, pages 260–261.
5
3
x2 x 2
2
5
5 2 3
5 2
2
x x
2
2
4
4
7
5 2 49
5
x
;x 4
4
16
4
5 7
5
7
x or x 4 4
4
4
1
x 3
x or
2
()
(
()
)
Chapter 10 255
For More Practice Go To:
Solve each equation by completing the square. Then check. Check students’ work.
11. 5x2 6 13x 0
6
13
x
5
5
13
13 2 6
13 2
2
x x
5
5
10
10
13 2
289
x
10
100
13 17
13
17
x
or x 10 10
10
10
2
x or
x 3
5
11
4
x
3
3
11
11 2 4
11 2
2
x x
3
3
6
6
11 2
169
x
6
36
11 13
11
13
x
or x 6
6
6
6
1
x or
x 4
3
x2 ( )
(
)
x2 ( )
( )
(
)
13. 15x2 16 34x
34
16
x
15
15
2
34
17
16
17
x2 x 15
15
15
15
17 2
49
x
15
225
17
7
17
7
or x x
15
15 15
15
8
2
x or
x
5
3
2
( )
2
3
2
x2 2x 1 1
3
1
2
(x 1) 3
3
x1
or x 1 3
3
3
3
x1
or x 1 3
3
3
17. 48x 36x2 11
1
27
x2 x 2
16
1
1 2 27
1 2
2
x x
16
2
4
4
1 2 7
x
4
4
1
1
7
x or x 7
4
4
2
2
1
1
7
or x 7
x 4
4
2
2
()
()
Solve.
19. Ten less than three times the square of a
number is 0. Which numbers are possible?
10
;
3
x 10 30 ; Possible numbers are 30
3
3
3
30
and .
3
31
9
31
x2 4x 4 4
9
5
2
(x 2) 9
5
x2
or x 2 5
3
3
5
x2
or x 2 5
3
3
x2 4x 18. 12x 7x2 5
4
11
x2 x 3
36
11
4
2 2
2 2
2
x x
36
3
3
3
2 2
5
x
3
36
2
2
5
x or x 5
3
3
6
6
2
2
5
or x 5
x 3
3
6
6
()
( )
21
25
21
2
x 2x 1 1
25
4
(x 1)2 25
2
2
x 1 or x 1 5
5
7
3
x or
x
5
5
x2 2x 15. 9x2 36x 31
x2 2x 16. 8x 16x2 27
()
( )
( )
14. 3x2 6x 2
x2 ( )
(
)
12. 25x2 21 50x
7x2 12x 5
12
5
x2 x 7
7
12
6 2
5
2
x x
7
7
7
6 2
1
x
7
49
6
6
1
x or x 7
7
7
6
1
x
or x 6 7
7
()
( )
(67)
2
1
7
1
20. The lengths of the sides of a square are increased
by 5 in. If the new area of the square is 39 in.2,
what was the original length of the square?
Let x the number; Solve: 3x2 10 0; x2 21. (123)(98)
(123)(100 2)
12,300 246 12,054
256 Chapter 10
Make a drawing; let x old length, so x 5 new
length; (x 5)2 39; x 5 39; The length
of the side cannot be negative, so the original
length is 5 39 inches.
22. (23.4)(0.9)
(23.4)(1 0.1)
23.4 2.34 21.06
23. 59.8 17.9 13.2 9.1
73 27 46
Copyright © by William H. Sadlier, Inc. All rights reserved.
10. 3x2 4 11x 0