Unit 1 Expressions, Equations and Inequalities 1.5 Quadratic Equations Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1 Objectives: • • • • Solve quadratic equations by factoring. Solve quadratic equations by the square root property. Solve quadratic equations using the quadratic formula. Solve equations reducible to quadratic form. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2 Definition of a Quadratic Equation A quadratic equation in x is an equation that can be written in the general form ax 2 bx c 0 where a, b, and c are real numbers, with a 0 A quadratic equation in x is also called a second-degree polynomial equation in x. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 3 The Zero-Product Principle To solve a quadratic equation by factoring, we apply the zero-product principle which states that: If the product of two algebraic expressions is zero, then at least one of the factors is equal to zero. If AB = 0, then A = 0 or B = 0. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 4 Solving a Quadratic Equation by Factoring 1. If necessary, rewrite the equation in the general form ax 2 bx c 0 , moving all nonzero terms to one side, thereby obtaining zero on the other side. 2. Factor completely. 3. Apply the zero-product principle, setting each factor containing a variable equal to zero. 4. Solve the equations in step 3. 5. Check the solutions in the original equation. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 5 Example: Solving Quadratic Equations by Factoring Solve by factoring: 2 x 2 x 1 Step 1 Move all nonzero terms to one side and obtain zero on the other side. 2 x2 x 1 0 Step 2 Factor (2 x 1)( x 1) 0 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 6 Example: Solving Quadratic Equations by Factoring (continued) Steps 3 and 4 Set each factor equal to zero and solve the resulting equations. (2 x 1)( x 1) 0 2x 1 0 2x 1 1 x 2 x 1 0 x 1 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 7 Example: Solving Quadratic Equations by Factoring (continued) Step 5 Check the solutions in the original equation. 2 x2 x 1 Check 1 x 1 x 2 2 2(1) 1 1? 2 1 1 2 1? 2 1 1 1 1 2 2 1 1 11 1 2 2 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 8 Solving Quadratic Equations by the Square Root Property Quadratic equations of the form u2 = d, where u is an algebraic expression and d is a nonzero real number, can be solved by the Square Root Property: If u is an algebraic expression and d is a nonzero real number, then u2 = d has exactly two solutions: u d u d or Equivalently, If u2 = d, then u d Copyright © 2014, 2010, 2007 Pearson Education, Inc. 9 Example: Solving Quadratic Equations by the Square Root Property Solve by the square root property: 5 x 45 0 2 5 x 2 45 x 2 9 x 9 x 3i Copyright © 2014, 2010, 2007 Pearson Education, Inc. 10 Example: Solve the equation: x 4x 1 0 2 x 2 5 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 11 The Quadratic Formula The solutions of a quadratic equation in general form ax 2 bx c 0 with a 0 , are given by the quadratic formula: b b 2 4ac x 2a Copyright © 2014, 2010, 2007 Pearson Education, Inc. 12 Example: Solving a Quadratic Equation Using the Quadratic Formula Solve using the quadratic formula: 2x 2x 1 0 a = 2, b = 2, c = – 1 2 b b 2 4ac x 2a (2) (2) 2 4(2)(1) x 2(2) Copyright © 2014, 2010, 2007 Pearson Education, Inc. 13 Example: Solving a Quadratic Equation Using the Quadratic Formula (continued) 2 4 8 x 4 2 12 x 4 x 2 1 3 4 1 3 x 2 2 2 3 x 4 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 14

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