SECTION A.6 Solving Equations A55 Solution Solving Equations by Factoring Solve the equation: − − + = x x x4 4 0 3 2 EXAMPLE 12 Do you recall the method of factoring by grouping? (If not, review Example 9(f) on p. A28.) Group the terms of − − + = x x x4 4 0 3 2 as follows: ( ) ( ) − − − = x x x4 4 0 3 2 Factor out x2 from the first grouping and 4 from the second. ( ) ( ) − − − = x x x 1 4 1 0 2 This reveals the common factor ( ) −x 1 , so ( )( ) ( )( )( ) − − = − + − = − = + = = = − − = = x x x x x x x x x x x 4 1 0 2 2 1 0 2 0 or 2 0 2 2 or 1 0 1 2 The solution set is { } −2, 1, 2 . Check: ( ) ( ) ( ) =− − −− −−+=−−++= = − −⋅+=−−+= = − −⋅+=−−+= x x x 2: 2 2 4 2 4 8 4 8 4 0 1:1 1 414 1144 0 2:2 2 424 8484 0 3 2 3 2 3 2 Factor again. Set each factor equal to 0. Solve. 2− is a solution. 1 is a solution. 2 is a solution. Now Work problem 91 ‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. A.6 Assess Your Understanding 1. Find the least common denominator of − − + x x x 3 4 and 5 3 2 . 2 2 (pp. A39–A40) 2. Factor − − x x 2 3. 2 (pp. A27–A28) 3. The solution set of the equation ( )( ) − + = x x 3 3 5 0 is ______. (p. A4) 4. True or False: = x x . 2 (pp. A9–A10) Concepts and Vocabulary 5. Two equations that have the same solution set are called . 6. An equation that is satisfied for every choice of the variable for which both sides are defined is called a(n) . 7. True or False: The solution of the equation − = x3 8 0 is 3 8 . 8. True or False: Some equations have no solution. 9. To solve the equation − = x x9 0 2 by completing the square, you would the number to both sides. 10. The quantity − b ac 4 2 is called the of a quadratic equation. If it is , the equation has no real solution. 11. True or False: Quadratic equations always have two real solutions. 12. True or False: If the discriminant of a quadratic equation is positive, then the equation has two solutions that are negatives of one another. 13. Multiple Choice A quadratic equation is sometimes called a equation. (a) first-degree (b) second-degree (c) third-degree (d) fourth-degree 14. Multiple Choice Which of the following quadratic equations is in standard form? (a) − = x x7 5 2 (b) = x 9 2 (c) ( )( ) + − = x x 5 4 0 (d) = − − x x 0 5 6 1 2 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure
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