Review 2015 – Solving Quadratics – ANSWER KEY
Solve each quadratic equation, using the method in the first column.
Method
Example/SOLVE
Strengths/Weaknesses
Graphing
Is accurate with rational
roots, not accurate with
irrational roots. Can be
very imprecise without a
graphing calculator.
When to use
When you have a
graphing utility
available.
Factoring
x 2  7 x  10  0
( x  2)( x  5)  0
x  2,5
One of the easiest ways
to solve. Not all
quadratics are factorable.

Square
Root
Property
 x  2
Can solve quadratics that When it is easy to set up
aren’t factorable. Can be as a perfect square.
difficult to convert to
square root.
Complete
the
Square
x 2  4 x  18  0
2
5
x  2  2.24, x  2  2.24
x  4.24, .24
x 2  4 x  4  18  4
( x  2) 2  22
x  2  4.69, x  2  4.69
x  6.69, 2.69
Quadratic
Formula
2 x2  5x  3  0
x
5  (5) 2  4(2)(3)
2(2)
If front coefficient is one
and it isn’t factorable, it
can be a good time to
use. It can get very ugly
quickly, especially with
a front coefficient
greater than 1.
Can be the only way to
solve a quadratic. Can
take a long time to solve.
WHENEVER
POSSIBLE!!!!
If you have front
coefficient of 1 and/or it
is already a perfect
square trinomial.
Anytime….but I would
try to factor first!
5 1
4
5 1
x
4
3
x  ,1
2
x
1
Solving Multiple Variations
Factoring
Quadratic Formula
2 y2  5 y  2  0
(2 y  1)( y  2)  0
1
y  , 2
2
2 y 2 + 5y + 2 = 0
3x 2  x  2  0
(3 x  2)( x  1)  0
2
x  , 1
3
3x 2  x  2  0
4 x 2  12 x  9  0
4 x 2  12 x  9  0
1
x=
-5 ± 25 - 4(2)(2)
4
-5 ± 9
4
-5 ± 3
x=
4
-1
x = ,-2
2
x=
2
1  1  4(3)(2)
6
1  25
x
6
1  5
x
6
2
x  , 1
3
x
3
(2 x  3) 2  0
3
x
2
12  144  4(4)(9)
8
12  0
x
8
3
x
2
x
4
2 y2  9 y  4  0
(2 y  1)( y  4)  0
1
y  , 4
2
2
2 y2  9 y  4  0
9  81  4(2)(4)
4
9  49
y
4
9  7
y
4
1
y  , 4
2
y
Completing the Square
Quadratic Formula
x 2 + 4x - 7 = 0
x 2 + 4x - 7 = 0
1
x 2 + 4x + 4 = 7 + 4
(x + 2) = 11
2
x + 2 = 3.32,x + 2 = -3.32
x = 1.32,-5.32
x=
-4 ± 16 - 4(1)(-7)
2
-4 ± 43
2
-4 ± 6.63
x=
2
x = 1.32,-5.32
x=
2
x 2 +18x + 74 = 0
x 2 +18x + 81 = -74 + 81
(x + 9) = 7
2
x + 9 = 2.65,x + 9 = -2.65
x = -6.35,-11.65
x 2 +18x + 74 = 0
x=
-18 ± 324 - 4(1)(74)
2
-18 ± 28
2
-18 ±5.29
x=
2
x = -6.35,-11.65
x=
3
x 2 + 4x -1 = 0
x 2 + 4x + 4 = 1+ 4
(x + 2) = 5
2
x + 2 = 2.24,x + 2 = -2.24
x = .24,-4.24
x 2 + 4x -1 = 0
x=
-4 ± 16 - 4(1)(-1)
2
-4 ± 20
2
-4 ± 4.47
x=
2
x = .24,-4.24
x=
3
4
x 2 -16x +57 = 0
x 2 -16x + 64 = -57 + 64
(x - 8) = 7
2
x - 8 = 2.65,x - 8 = -2.65
x = 10.65,5.35
x 2 -16x +57 = 0
x=
16 ± 256 - 4(1)(57)
2
16 ± 28
2
16 ±5.29
x=
2
x = 10.65,5.35
x=
Solving – Sorting Challeng
Square Root Property
( x -1)
2
=8
1. x -1 = 2.83,x -1 = -2.83
(
)
2
2. x - 2 = -4
NO SOLUTION!!!!
x = 3.83,-1.83
( x + 4)
2
= 25
3. x + 4 = 5,x + 4 = -5
x = 1,-9
( x + 3)
2
=5
4. x + 3 = 2.24,x + 3 = -2.24
x = -0.76,-5.24
Factoring
x 2 + 6x + 8 = 0
5. (x + 2)(x + 4) = 0
x = -2,-4
x 2 -11x +10 = 0
7. (x -10)(x -1) = 0
x = 10,1
x 2 -10x + 9 = 0
6. (x - 9)(x -1) = 0
x = 9,1
4x 2 + 4x +1 = 0
8. (2x +1)2 = 0
-1
x=
2
Complete the Square
x 2 -10x + 8 = 0
x 2 -10x + 25 = -8 + 25
2
9. (x -5) = 17
x -5 = 4.12,x -5 = -4.12
x 2 + 4x + 7 = 0
2
10. x + 4x + 4 = -7 + 4
(x + 2)2 = -3
NO SOLUTION!
x = 9.12,0.88
4
11.
x 2 - 2x - 9 = 0
x 2 + 6x - 3 = 0
x 2 - 2x +1 = 9+1
x 2 + 6x + 9 = 3+ 9
(x -1)2 = 10
x -1 = 3.16,x -1 = -3.16
x = 4.16,-2.16
2
12. (x + 3) = 12
x + 3 = 3.46,x + 3 = -3.46
x = .46,-6.46
Quadratic Formula
5x 2 + 8x -1 = 0
x=
-8 ± 64 - 4(5)(-1)
10
13. x = -8 ± 84
10
-8 ± 9.17
x=
10
x = .12,-1.72
x 2 + 3x +5 = 0
15. x =
-3± 9- 4(1)(5)
2
-3± -11
2
NO SOLUTION!
x=
2x 2 - 4x +1 = 0
x=
4 ± 16 - 4(2)(1)
4
14. x = 4 ± 8
4
4 ± 2.83
x=
4
x = .29,
x 2 + 3x - 3 = 0
x=
-3± 9- 4(1)(-3)
2
16. x = -3± 21
2
-3± 4.58
x=
2
x = 0.79,-3.79
Solving – YOUR CHOICE 
You will NOT have a graphing calculator on your test!
x 2  3x  4
1.
x 2  3x  4  0
( x  4)( x  1)  0
x  4, 1
x2  5  3
3. x  8
x  2.83
2
x 2  10x  25  0
2
2. ( x  5)  0
x 5
 x  4
2
8
4. x  4  2.83, x  4  2.83
x  1.17, 6.83
5
x2  6 x  8  0
5. ( x  4)( x  2)  0
x  4, 2
x 2  3x  1  6
x 2  3x  5  0
3  9  4(1)(5)
2
7.
3  29
x
2
3  5.39
x
2
x  4.195, 1.195
x
8 x 2  200
2
9. x  25
x  5
3x 2  7 x  2  0
6. (3 x  1)( x  2)  0
1
x  ,2
3
4 x2  7 x  2  0
7  49  4(4)(2)
8
7  32
8. x 
8
7  4.12
x
8
x  0.36, 1.39
x
4 x2  6 x 1  0
6  36  4(4)(1)
8
6  52
10. x 
8
6  7.21
x
8
x  1.65, .15
x
6