 # 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:

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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 4x 2 + 20x -12

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

Factor Diamonds x² + 8x + 7 =0 = (x + 1) (x + 7) = 0 7 8 17 So your answers are -1 and -7

Practice: Solve by a factor diamond X 2 + 15x + 36 (x+3)(x+12)

Bottom’s up (Borrowing Method) 12 13 112 2x² + 13x + 6 =0 x² + 13x + 12 =0 = (x + 12) (x + 1) =0 2 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 2x 2 – 19x + 24 (x-8)(2x-3)

Factor by Grouping -30 -7 3-10 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 3x 2 + 7x - 20 (x+4)(3x-5)

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

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

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