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 y = (x + 2)(x – 4) By looking at the roots, we can get the solutions. Here, the solutions are -2 and 4.

Golden Rules of Factoring

Example: Factor out the greatest common factor 4x2 + 20x -12

Practice: Factor each expression Solutions: a.) 3(3x2 + x – 6) b) 7(p2 + 3) c) 2w(2w + 1) a) 9x2 + 3x – 18 b) 7p2 + 21 c) 4w2 + 2w

Factor Diamonds x² + 8x + 7 =0 = (x + 1) (x + 7) = 0 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) 2x² + 13x + 6 =0 x² + 13x + 12 =0 12 12 1 = (x + 12) (x + 1) =0 13 2 2 = (x + 6) (x + 1) =0 2 Multiply by 2 to get rid of the fraction = (x + 6) (2x + 1) =0 So your answers are -6 and -1/2

Practice: Solve using Bottom’s Up/Barrowing Method 2x2 – 19x + 24 (x-8)(2x-3)

Factor by Grouping 2x² – 7x – 15 =0 2x² – 10x + 3x – 15 =0 -30 -10 3 -7 Note: you are on the right track because you have (x-5) in both parenthesis 2x(x – 5) + 3(x – 5) =0 (2x + 3)(x – 5)=0 So your answers are -3/2 and 5

Practice: Factor by Grouping 3x2 + 7x - 20 (x+4)(3x-5)

SHORTCUTS a2 + 2ab + b2 (a+b)2 a2 - 2ab + b2 (a - b)2 Example: 25x2 + 90x + 81 (5x + 9)2 a2 - 2ab + b2 (a - b)2 Example: 9x2 – 42x + 49 (3x – 7)2 a2 - b2 (a+b)(a - b) Example: x2 – 64 (x + 8)(x – 8)

Practice Problems: Solve using any method Solutions: a) (x-6)(3x+2) (x+2)(4x-3) (2x+7)(2x-7) (x+4)(2x+3) 3x2 – 16x – 12 4x2 + 5x – 6 4x2 – 49 2x2 + 11X + 12