132 CHAPTER 1 Equations and Inequalities EXAMPLE 1 Using the Zero-Factor Property Solve 6x2 + 7x = 3. SOLUTION 6x2 + 7x = 3 6x2 + 7x - 3 = 0 Standard form 13x - 1212x + 32 = 0 Factor. 3x - 1 = 0 or 2x + 3 = 0 Zero-factor property 3x = 1 or 2x = -3 Solve each equation. x = 1 3 or x = - 3 2 CHECK 6x2 + 7x = 3 Original equation Don’t factor out x here. 6a 1 3b 2 + 7a 1 3b ≟3 Let x = 1 3 . 6 9 + 7 3 ≟3 3 = 3 ✓ True 6a3 2b 2 + 7a3 2b ≟3 Let x = -3 2 . 54 4 - 21 2 ≟3 3 = 3 ✓ True Both values check because true statements result. The solution set is E -3 2 , 1 3F. S Now Try Exercise 15. The Square Root Property When a quadratic equation can be written in the form x2 = k, where k is a constant, the equation can be solved as follows. x2 = k x2 - k = 0 Subtract k. A x - 2k B A x + 2k B = 0 Factor. x - 2k = 0 or x + 2k = 0 Zero-factor property x = 2k or x = -2k Solve each equation. This proves the square root property. Square Root Property If x2 = k, then x =!k or x = −!k. That is, the solution set of x2 = k is E ! k, −!k F , which may be abbreviated E t!k F . Both solutions 2k and -2k of x2 = k are real if k 70. Both are pure imaginary if k 60. If k 60, then we write the solution set as E ti !∣ k ∣ F . If k = 0, then there is only one distinct solution, 0, sometimes called a double solution.
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