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REVIEW OF SOLVING ALGEBRAIC EQUATIONS Solving by Factoring: a) x2 + 6x – 16 = 0 Look for two numbers that multiply to be –16 and add to be 6. –16 and 1 –8 and 2 –4 and 4 –1 and 16 –2 and 8 x2 + 6x – 16 = 0 (x – 2)(x + 8) = 0 So, either x – 2 = 0 or x = 2 or x+8 = 0 x = –8 b) 2x2 – x – 3 = 0 This will factor into something of the form: (2x ? )(x ? ) Use trial and error with factors of –3. (2x – 3)(x + 1) (2x + 1)(x – 3) (2x + 3)(x – 1) (2x – 1)(x + 3) = = = = 2x2 + 2x – 3x – 3 = 2x2 – x – 3 2x2 – 6x + x – 3 = 2x2 – 5x – 3 2x2 – 2x +3x – 3 = 2x2 + x – 3 2x2 + 6x – x – 3 = 2x2 + 5x – 3 Yes No No No 2x2 – x – 3 = 0 (2x – 3)(x + 1) = 0 So, either 2x – 3 = 0 x = 3/2 or x + 1 = 0 or x = –1 c) 9x2 – 6x + 1 = 0 This will factor into something of the form: (9x ? )(x ? ) Use trial and error with factors of 1. or (3x ? )(3x ? ) (9x + 1)(x + 1) = 9x2 + 9x + x + 1 = 9x2 + 10x + 1 No (9x – 1)(x – 1) = 9x2 – 9x – x + 1 = 9x2 – 10x + 1 No (3x + 1)(3x + 1) = 9x2 + 3x +3x + 1 = 9x2 + 6x + 1 No (3x – 1)(3x – 3) = 9x2 – 3x – 3x + 1 = 9x2 – 6x + 1 Yes 9x2 – 6x + 1 = 0 (3x – 1)(3x – 1) = 0 So, 3x – 1 = 0 x = 1/3 d) 4x2 – 64 = 0 4x2 – 64 = 0 4(x2 – 16) = 0 So, (factor out a 4) x2 – 16 = 0 (x + 4)(x – 4) = 0 (using the difference of squares formula) So, either x + 4 = 0 x=–4 or or x–4 = 0 x=4 e) 3x2 – 5x = 0 3x2 – 5x = 0 x(3x – 5) = 0 So, either x = 0 x=0 (factor out an x) or or 3x – 5 = 0 x = 5 /3 Solving using Square Roots: a) x2 – 49 = 0 x2 – 49 = 0 x2 = 49 x 2 49 x=±7 (square root both sides) b) 5x2 – 10 = 0 5x2 – 10 = 0 5x2 = 10 x2 = 2 x2 2 x=± 2 (square root both sides) c) (3x + 5)2 = 8 (3x + 5)2 = 8 (3 x 5) 2 8 (square root both sides) 3x 5 8 3x 8 5 x 8 5 5 8 5 2 2 3 3 3 Solving by Completing the Square: Steps: 1. Move c over 2. Take half of b, square it, add that number to both sides 3. Factor and solve Note: If a ≠ 1, you must divide everything by a first. a) x2 – 6x – 11 = 0 x2 – 6x – 11 = 0 x2 – 6x + _____ = 11 (move over the 11 and leave a space) x2 – 6x + __9__ = 11 + 9 (half of 6 is 3. 3 squared is 9. Add 9 to both sides) x2 – 6x + 9 = 20 (x – 3)(x – 3) = 20 (left side should always factor into “twin” factors) (x – 3) 2 = 20 x 3 20 x 20 3 3 20 3 2 5 b) 2x2 + 3x – 4 = 0 2 x 2 3x 4 0 2 2 3 4 x x 0 2 2 2 2 3 x 2x2 0 (divide everything by 2 first) x 2 32 x _____ 2 (move the 2 over) x 2 32 x 9 16 2 169 x 2 32 x 9 16 ( x 34 ) 2 x 3 4 x 3 4 x 41 4 41 16 41 16 41 16 41 4 43 3 41 4 (half of 3 2 = 1 2 32 34 . 34 2 196 . Add 9 16 to both sides.)