Beginning Algebra 5.7 Solving Equations by Factoring:

6.6 Solving Equations by Factoring: 1. To solve an equation by writing it in standard form and then factoring.

6.6 Solving Equations by Factoring: is the standard form of a quadratic equation where a, b, and c are real numbers numbers (a > 0). For real numbers numbers A and B, if A·B A·B = 0 then A = 0 or B = 0, or both both. Zero-Factor Theorem: ax 2 + bx + c = 0

Solve the equation (x – 3)(x – 2) = 0 What to do How to do it Solve: (x – 3)(x – 2) = 0 Zero-Factor Theorem: Solve each each Factor = 0 Check first first Solution x = 3 and x = 2 (x – 3) = 0, (x – 2) = 0 [(3) – 3][(3) – 2] = 0 Check second second Solution [(2) – 3][(2) – 2] = 0 Solutions: (x – 3)(x – 2) = 0 x = 3 and x = 2 S = {2, 3} Solution: S = {2, 3}

Solve the equation Factor left side side. x 2 x 2 – 5x – 6 = 0 What to do How to do it (x – 6)(x + 1) = 0 Solve: x 2 – 5x – 6 = 0 Find numbers with product product 6:6: difference difference 55 55, larger sign sign – Set each each Factor = 0. (x – 6) = 0, (x + 1) = 0 Solve each each Factor = 0. Check each Solution. x = 6 and x = - 1 (6) 2 – 5(6) – 6 = 0 and (-1) 2 – 5(-1) – 6 = 0 GN = 6  - 6, +1 S = {6, -1} Solution: S = {6, -1}

Solve the equation equation: Factor difference of squares y 2 = 324 What to do How to do it (y + 18)(y – 18) = 0 Solve: y 2 = 384 Rearrange the polynomial polynomial: y 2 – 324 = 0 Set each each Factor = 0. (y + 18) = 0, (y – 18) = 0 Solve each each Factor = 0. Check each each Solution. y = -18 and y = 18 (-18) 2 = 384 and (18) 2 = 384 S = {-18, 18} Solution: S = {-18, 18}

Solve the equation equation: Factor the left side side. x 2 = 7x – 12 What to do How to do it (x – 4)(x – 3) = 0 Solve: x 2 = 7x – 12 Rearrange the polynomial polynomial: x 2 – 7x + 12 = 0 Set each each Factor = 0. (x – 4) = 0, (x – 3) = 0 Solve each each Factor = 0. Check each each Solution. x = 4 and x = 3 (4) (4) 2 = 7 (4) (4) – 12 and (3) (3) 2 = 7 (3) – 12 S = {3, 4} Solution: S = {3, 4}

Solve the equation equation: Factor the left side side. z 2 = 12z What to do How to do it z(z – 12) = 0 Solve: z 2 = 12z Rearrange the polynomial polynomial: z 2 – 12z = 0 Set each each Factor = 0. z = 0, (z – 12) = 0 Solve each each Factor = 0. Check each each Solution. z = 0 and z = 12 (0) (0) 2 = 12(0) and (12) 2 = 12(12) S = {0, 12} Solution: S = {0, 12}

Solve the equation equation: Factor the left side side. 6t 2 – t = 15 What to do How to do it (2t + 3)(3t – 5) = 0 Solve: 6t 2 – t = 15 Rearrange the polynomial polynomial: 6t 2 – t – 15 = 0 Set each each Factor = 0. (2t + 3) = 0, (3t – 5) = 0 Solve each each Factor = 0. Check each each Solution. t = -3 2 t = 3 5 and = = 15 - and S = Solution: S = ,

THE END 5.7