Solving Quadratic Equations
Solving quadratic equations (equations with x2 can be done in different ways. We will use two
different methods. What both methods have in common is that the equation has to be set to = 0.
For instance, if the equation was x2 – 22 = 9x, you would have to subtract 9x from both sides of
the equal sign so the equation would be x2 – 9x – 22 = 0.
Solve by factoring: After the equation is set equal to 0, you factor the trinomial.
x2 – 9x – 22 = 0
(x-11) (x+2) = 0
Now you would set each factor equal to zero and solve. Think about it, if the product of the two
binomials equals zero, well then one of the factors has to be zero.
x2 – 9x – 22 = 0
(x-11) (x+2) = 0
x – 11 = 0
x+2=0
+11 +11
-2
x = 11
x = -2
or
-2
* Check in the ORIGINAL equation!
Solving Quadratics by Factoring:
20) x2 - 5x - 14 = 0
21) x2 + 11x = -30
22) x2 - 45 = 4x
23) x2 = 15x - 56
24) 3x2 + 9x = 54
25) x3 = x2 + 12x
26) 25x2 = 5x3 + 30x
27) 108x = 12x2 + 216
29) 10x2 - 5x + 11 = 9x2 + x + 83
28) 3x2 - 2x - 8 = 2x2
30) 4x2 + 3x - 12 = 6x2 - 7x – 60
1
Solve each quadratic by factoring:
1) x2 – 3x – 108 = 0
2) x2 – 16x = - 55
3) x2 + 28 = 29x
4) x2 = x + 30
5) 6x2 = 54x – 120
6) x3 – 54x = 3x2
2
7) 7x2 + 14x = 560
8) 4x3 = -52x2 – 168x
9) 12x2 – 48x + 96 = 96
10) 15x3 – 28x = 105x2 – 28x
11) 4x2 – 96x = - 576
12) x2 – x = 5,256
3
Solve using the quadratic formula:
When ax2 + bx + c = 0
x=
-b ± √b2 – 4ac .
2a
a is the coefficient of x2 b is the coefficient of x c is the number (third term)
Notice the ±
is what will give your two answers (just like you had when solving by factoring)
x2 – 9x – 22 = 0
a=1
x=
-b ± √b2 – 4ac .
2a
b= - 9
c = -22
x=
-(-9) ± √ (-9)2 – 4(1)(-22)
2(1)
x=
9 ± √81 + 88
2
x=
9 ± √169 .
2
-4(1)(-22) = 88
Split and do the + side and - side
9 – 13
2
9 + 13
2
x = 11
or
x = -2
* Check in the ORIGINAL equation!
Solving Quadratics Using the Quadratic Formula:
31) 2x2 - 6x + 1 = 0
32) 3x2 + 2x = 3
33) 4x2 + 2 = -7x
34) 7x2 = 3x + 2
35) 3x2 + 6 = 5x
36) 9x - 3 = 4x2
4
Proportions and Percents
Proportions:
A proportion is a statement that two ratios are equal. When trying to solve proportions we use
the Cross Products Property of Proportions.
A = C
A(D) = B(C)
B
D
Example:
6__ = x__
x + 5__ =
1.5___
11
121
12
6
6(121) = 11x
6(x + 5) = 12(1.5)
726 = 11x
6x + 30 = 18
-30
-30
6x
= -12
6
6
x
= -2
726 = 11x
11
11
66 = x
1)
x
14
_
=
16
35
2)
x–3 _ =
x+3
12
30
_
Percents:
Is
=
%___
Of
100
Example:
What number is 20% of 50?
Is:
?x
x
=
20 .
Of:
of 50
50
100
%:
20%
100: 100
100x = 20(50)
100x = 1,000
100x = 1,000
100
100
x = 10
a) What number is 40% of 160?
b) 48 is what percent of 128?
c) 28 is 75% of what number?
d) What number is 36% of 400?
5
Part I:
1)
x . =
12
18
54
.
2) -
13
x
. =
65
90
.
3) x + 4 . =
9
4) - 16 . =
8 .
6x-2
11
6) What is 20% of 32?
5)
14
. =
16
7) 72 is 40% of what number?
8) 21.56 is what percent of 98?
9) - 31 is what percent of -124?
6x .
18
3x
.
3x + 3
10) What is 62% of 140?
Part II:
1)
x . =
12
13
78
4)
- 16 . =
5x-2
8 .
11
.
2)
- 13
x
5)
x+5
x-3
. =
195
150
. =
x
9
.
.
3) x + 4 . =
9
6x .
18
6)
9 _
x+8
x-4 _ =
12
7) 12 is 40% of what number?
8) 21.56 is what percent of 98?
9) 45 is what percent of 180?
10) What is 62% of 70?
Part III:
1)
23
x
4)
x+1
x+6
.
=
.
57.5
45
=
.
2
x
.
2) 3x – 5 . =
13
5x + 1 .
52
5) 2x – 4 . =
x+5
x-2 .
x+1
10) What is 80% of 850?
8) 128 is 32% of what number?
9) 72 is what percent of 120?
10) What is 80% of 850?
3) 5x -1
10x+5
6) x + 7
2x – 1
= 33 .
45
= x+6 .
x-2
6
Mixed Equations: Figure out what type of equation you have and then pick a strategy
to solve.
1) 20 - (5/8)x = 40
2) 6(7x - 2) = 8(4x + 1)
3) 2(5x - 4) - 3(4x + 3) = -43
4) x2 + 44 = 15x
5) 3x2 + 18x = 81
6) 3x2 = 5x + 5
7) 11x - 5 = 7x - 53
8) 6(3x + 1) + 5(10 - 4x)= 39
9) ¼x - 33 = -49
10) 7x2 - 1 = 3x
11) 9(3x + 1) = 8(5x + 6)
12) 15x = x2 – 16
13) x2 + 8x = 12
14) 9(4x + 7) - 6(7x + 10) = -54
15) 44 = 20 - 2x
16) 4x2 - 128 = 16x
17) 3x2 - 8x + 6 = x + 6
18) 7(6x + 2) = 10(3x + 5)
19) 3x2 + 13x - 12 = 9x2 - 11x - 12
21)
24)
14 . =
8x - 4
10 . =
7x + 2
35
50
8 .
5x + 4
.
22)
25)
20) 2x2 - 14 = 10x
x+5
x-4
. =
x-6
. =
2x - 3
x
32
.
x + 12 .
x+4
23) x - 10_ =
12
6 _
x-4
26) 2x - 3 =
x+1
x-3 _
x+3
7