Mat0024 l8-16-sections 11-6-7

11.6 – Solving Quadratic Equations by Factoring
A quadratic equation is written in the Standard
Form, 2
0ax bx c+ + =
where a, b, and c are real numbers and .0a ≠
Examples:
2
7 12 0x x− + =
2
3 4 15x x+ =
( )7 0x x + =
(standard form)

Zero Factor Property:
If a and b are real numbers and if ,0ab =
Examples:
( )7 0x x + =
then or .0a = 0b =
0x = 7 0x + =
7x = −0x =

Zero Factor Property:
If a and b are real numbers and if ,0ab =
Examples:
( ) ( )10 3 6 0x x− − =
then or .0a = 0b =
10 0x − = 3 6 0x − =
10x =
3 6x = 2x =
10 10 01 0x − =+ + 63 66 0x − + = +
3 6
3 3
x
=

Solving Quadratic Equations:
1) Write the equation in standard form.
4) Solve each equation.
2) Factor the equation completely.
3) Set each factor equal to 0.
5) Check the solutions (in original equation).

2
3 18 0x x− − =
6 0x − =3 0x + =
( )3x +
6x =3x = −
2
3 18x x− =
18:Factors of
1,18 2, 9 3, 6
( ) ( )
2
6 3 16 8− =
36 18 18− =
18 18=
( ) ( )
2
13 3 83− −− =
9 9 18+ =
18 18=
( )6x − 0=

( )3 18x x − =
18x =
2
3 18x x− =
( ) ( )
2
18 13 18 8− =
324 54 18− =
270 18≠
( ) ( )
2
21 23 11 8− =
441 63 18− =
378 18≠
3 18x − =
3 183 3x − =+ +
21x =
If the Zero Factor
Property is not used,
then the solutions will
be incorrect

2
4 5x x− =
1 0x + = 5 0x − =
( ) ( )1 5 0x x+ − =
1x = − 5x =
( )4 5x x − =
2
4 5 0x x− − =

2
3 7 6x x+ = 3 0x + = 3 2 0x − =
( ) ( )3 3 2 0x x+ − =
3x = −
2
3
x =
( )3 7 6x x + =
2
3 7 6 0x x+ − = 3 2x =
6:Factors of
2, 31, 6
3:Factors of
1, 3

2
9 24 16x x− = −
2
9 24 16 0x x− + =
3 4 0x − =
( ) ( )3 4 3 4 0x x− − =
4
3
x =
3 4x =
( )9 16and are perfect squares

3
2 18 0x x− =
2x
2 0x =
2x
3x = −
3 0x + = 3 0x − =
3x =0x =
( )2
9x − 0=
( )3x + ( )3x − 0=

( ) ( )2
3 3 20 7 0x x x+ − − =
( )3x +
3 0x + =
7x =
7 0x − = 3 1 0x + =
1
3
x = −
3x = − 3 1x = −
3:Factors of 1, 3 7 :Factors of 1, 7
( )7x − 0=( )3 1x +

0 =
A cliff diver is 64 feet above the surface of the
water. The formula for calculating the height (h)
of the diver after t seconds is: 2
16 64.h t= − +
How long does it take for the diver to hit the
surface of the water?
0 =
0 =
2 0t + = 2 0t − =
2t = − 2t = seconds
2
16 64t− +
16− ( )2
4t −
16− ( )2t + ( )2t −
11.7 – Quadratic Equations and Problem Solving

2
x
The square of a number minus twice the number is
63. Find the number.
( )7x +
7x = −
x is the number.
2
2 63 0x x− − =
7 0x + = 9 0x − =
9x =
2x− 63=
63:Factors of 1, 63 3, 21 7, 9
( )9x − 0=

( )5 176w w+ =
The length of a rectangular garden is 5 feet more than
its width. The area of the garden is 176 square feet.
What are the length and the width of the garden?
( )11w−
The width is w.
11 0w− =
11w =
The length is w+5.l w A× =
2
5 176w w+ =
2
5 176 0w w+ − =
16 0w+ =
16w = −
11w = 11 5l = +
16l =
feet
feet
176:Factors of
1,176 2, 88 4, 44
8, 22 11,16
( )16w+ 0=

x
Find two consecutive odd numbers whose product is
23 more than their sum?
Consecutive odd numbers: x
5x = − 5x =2
2 2 25x x x+ = +
2
25 0x − =
( )5x +
5 0x + = 5 0x − =
5, 3− − 5, 7
5 2 3− + = − 5 2 7+ =
2.x +
( )2x + = ( )2x x+ + 23+
2
22 2 2 25xx x x x+ = −+−
2
25 2525x =− −
2
25x =
( )5x − 0=

a x=
The length of one leg of a right triangle is 7 meters less than
the length of the other leg. The length of the hypotenuse is 13
meters. What are the lengths of the legs?
12a =
( ).Pythagorean Th
( )
22 2
7 13x x+ − =
5x = −
5=
meters
7b x= − 13c =
2 2
14 49 169x x x+ − + =
2
2 14 120 0x x− − =
( )2
2 7 60 0x x− − =
2
5 0x + = 12 0x − =
12x =
12 7b = − meters
2 2 2
a b c+ =
60:Factors of 1, 60 2, 30
3, 20 4,15 5,12
( )5x + ( )12x − 0=
6,10

Mat0024 l8-16-sections 11-6-7

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