Read the explanations, follow the examples, take notes, and complete the You Trys. Earlier, we learned that a linear equation in one variable is an equation that can be written in the form ax b 0, where a and b are real numbers and a 0. In this section, we will learn how to solve quadratic equations. Definition A quadratic equation can be written in the form ax2 bx c 0 where a, b, and c are real numbers and a 0. We say that a quadratic equation written in the form ax2 bx c 0 is in standard form. But quadratic equations can be written in other forms, too. Some examples of quadratic equations are x2 13x 36 0, 5n(n 3) 0, and (z 4)(z 7) 10. Quadratic equations are also called second-degree equations because the highest power on the variable is 2. There are many different ways to solve quadratic equations. In this section, we will learn how to solve them by factoring; other methods will be discussed later in this book. Solving a quadratic equation by factoring is based on the zero product rule: If the product of two quantities is zero, then one or both of the quantities is zero. For example, if 5y 0, then y 0. If p 4 0, then p 0. If ab 0, then either a 0, b 0, or both a and b equal zero. Definition Zero product rule: If ab 0, then a 0 or b 0. We will use this idea to solve quadratic equations by factoring. 1 Solve a Quadratic Equation of the Form ab 0 EXAMPLE 1 In-Class Example 1 Solve. a) y(y 10) 0 b) (4t 1)(t 5) 0 Answer: a) {0, 10} b) e 1 4 , 5 f Solve. a) p( p 4) 0 b) (3x 1)(x 7) 0 Solution a) The zero product rule says that at least one of the factors on the left must equal zero in order for the product to equal zero. p( p 4) 0 b R p 0 or p 4 0 Set each factor equal to 0. p 4 Solve. Check the solutions in the original equation: If p 0: If p 4: 0(0 4) 0 4(4 4) 0 0(4) 0 ✓ 4(0) 0 ✓ The solution set is {4, 0}. Where have you seen this last step before? Solving linear equations! 424 CHAPTER 7 Factoring Polynomials www.mhhe.com/messersmith
messersmith_power_introductory_algebra_1e_ch4_7_10
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