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)