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Solving Equations 5.7 S KM & PP 1 Factors that multiply to Zero ... ab=0 That’s Whatright! if ab = 0equal ? a or b could zero! Then a could equal zero! 5.7 S ...or b could equal zero! KM & PP 2 ZERO Product Rule If a a product b is zero, 0 then one or or morebofthe 0 a factors 0 must be zero! 5.7 S KM & PP 3 Zero Factor Property Solve: 2x = 0 2x 0 20 x0 0 5.7 S KM & PP 4 Zero Factor Property Solve: 2(x + 5) = 0 2x 5 0 20 x5 0 x 5 5 5.7 S KM & PP 5 Zero Factor Property Solve: 2x(x + 5) = 0 2xx 5 0 20 x 0 x5 0 x 5 0,5 5.7 S KM & PP 6 Zero Factor Property Solve: 2x(x + 5)(x-3) = 0 2xx 5 x 3 0 2 0 x 0 x5 0 x3 0 x 5 x 3 0,5,3 5.7 S KM & PP 7 Zero Factor Property Solve: (2x - 7)(4x + 3)= 0 2x 74x 3 0 2x 7 0 4x 3 0 4x 3 3 x 4 2x 7 7 x 2 7 3 , 2 4 5.7 S KM & PP 8 Zero Factor Property Solve: 6x2 = 3x Use the properties of equality to rearrange the terms of the equation so that it is equal to ZERO. 6x 3x 2 6x 3x 0 3x2x 1 0 2 30 x0 2x 1 0 2x 1 1 0, 2 5.7 S KM & PP 1 x 2 9 General Strategy for Solving Equations Using The Zero Factor Property 1) Arrange the equation so that one side is zero. 2) Completely factor the other side. 3) Set each factor equal to zero and solve, if possible. 4) Write the solution set. 5) Check each solution by substitution. 5.7 S KM & PP 10 Solve: x2 = 169 1) Arrange the equation so that one side is zero. x 169 0 2 2) Completely factor the expression. (x 13)(x 13) 0 3) Set each factor equal to zero and solve. x 13 0 x 13 or or x 13 0 x 13 4) Write the solution set. 13,13 5) Check each solution by substitution. ( 13)2 169 5.7 S (13)2 169 KM & PP 11 Solve: x2 + 25 = 10x 1) Arrange the equation so that one side is zero. x 10x 25 0 2 2) Completely factor the expression. (x 5)(x 5) 0 3) Set each factor equal to zero and solve. x 5 0 or x 5 0 x5 or x5 4) Write the solution set. 5 5) Check each solution by substitution. (5)2 25 105 50 50 5.7 S KM & PP 12 Solve: 3x2 = 2 - x 1) Arrange the equation so that one side is zero. 3x x 2 0 2 2) Completely factor the expression. (3x 2)(x 1) 0 3) Set each factor equal to zero and solve. 3x 2 0 2 x 3 or x 1 0 or x 1 4) Write the solution set. 2 ,1 3 5.7 S KM & PP 13 Solve: 3x2 = 2 – x continued 2 ,1 3 5) Check each solution by substitution. 2 2 2 3 2 3 3 4 6 2 3 9 3 3 3 1 2 1 2 31 2 1 33 4 4 3 3 5.7 S KM & PP 14 Solve: x + 12 = x(x – 3) 1) Arrange the equation so that one side is zero. x 12 x 2 3x x 4x 12 0 2 2) Completely factor the expression. (x 6)(x 2) 0 3) Set each factor equal to zero and solve. x 6 0 or x 2 0 x6 or x 2 4) Write the solution set. 6,2 5.7 S KM & PP 15 Solve: x + 12 = x(x – 3) continued 6,2 5) Check each solution by substitution. 6 12 66 3 5.7 S 2 12 2(2 3) 18 6(3) 10 2 5 18 18 10 10 KM & PP 16 Solve: 2x3 + 3x2 = 18x + 27 1) Arrange the equation so that one side is zero. 2x 3x 18x 27 0 3 2 2) Completely factor the expression. x (2x 3) 9(2x 3) 0 2 (x 9)(2x 3) 0 2 (x 3)(x 3)(2x 3) 0 3) Set each factor equal to zero and solve. x 3 0 x 3 0 2x 3 0 x 3 5.7 S or x3 KM & PP or 3 x 2 17 Solve: 2x3 + 3x2 = 18x + 27 continued 4) Write the solution set. 3 3,3, 2 5) Check each solution by substitution. 2 3 3 3 18 3 27 3 2 2 27 39 54 27 54 27 27 27 27 233 332 183 27 227 39 54 27 54 27 81 81 81 5.7 S KM & PP 18 Solve: 2x3 + 3x2 = 18x + 27 continued 4) Write the solution set. 3 3,3, 2 5) Check each solution by substitution. 3 2 3 3 3 2 3 18 27 2 2 2 27 9 2 3 27 27 8 4 27 27 0 4 4 00 5.7 S KM & PP 19 That’s All for Now! 5.7 S KM & PP 20