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5.4 Factoring Quadratic Expressions
WAYS TO SOLVE A QUADRATIC EQUATION ax² + bx + c = 0
• There are many ways to solve a quadratic.
• The main ones are:– Graphing– Factoring– Bottom’s Up– Grouping– Quadratic formula– Completing the square
By Graphing
By looking at the roots, we can get the solutions.Here, the solutions are -2 and 4.
y = (x + 2)(x – 4)
Golden Rules of Factoring
Example: Factor out the greatest common factor
• 4x2 + 20x -12
Practice: Factor each expression
a) 9x2 + 3x – 18
b) 7p2 + 21
c) 4w2 + 2w
Solutions:
a.) 3(3x2 + x – 6)
b) 7(p2 + 3)
c) 2w(2w + 1)
Factor Diamonds
x² + 8x + 7 =0
= (x + 1) (x + 7) = 0
7
817
So your answers are -1 and -7
Practice: Solve by a factor diamond
• X2 + 15x + 36
(x+3)(x+12)
Bottom’s up (Borrowing Method)
12
13112
2x² + 13x + 6 =0x² + 13x + 12 =0
= (x + 12) (x + 1) =02 2
= (x + 6) (x + 1) =0 2
So your answers are -6 and -1/2
Multiply by 2 to get rid of the fraction
= (x + 6) (2x + 1) =0
Practice: Solve using Bottom’s Up/Barrowing Method
• 2x2 – 19x + 24
(x-8)(2x-3)
Factor by Grouping
-30
-73-10
2x² – 7x – 15 =0
2x² – 10x + 3x – 15 =0
2x(x – 5) + 3(x – 5) =0
(2x + 3)(x – 5)=0
So your answers are -3/2 and 5
Note: you are on the right track because you have (x-5) in both parenthesis
Practice: Factor by Grouping
3x2 + 7x - 20
(x+4)(3x-5)
SHORTCUTS
• a2 + 2ab + b2 (a+b)2
Example: 9x2 – 42x + 49 (3x – 7)2
Example: 25x2 + 90x + 81 (5x + 9)2
• a2 - 2ab + b2 (a - b)2
• a2 - b2 (a+b)(a - b)
Example: x2 – 64 (x + 8)(x – 8)
Practice Problems: Solve using any method
a) 3x2 – 16x – 12
b) 4x2 + 5x – 6
c) 4x2 – 49
d) 2x2 + 11X + 12
Solutions:
a)(x-6)(3x+2)
b)(x+2)(4x-3)
c) (2x+7)(2x-7)
d)(x+4)(2x+3)