A54 APPENDIX Review The idea should now be clear. If an equation contains an expression and that same expression squared, make a substitution for the expression. You may get a quadratic equation. Now Work problem 87 Solve an Equation Quadratic in Form Find the real solutions of the equation: ( ) ( ) + + + − = x x 2 11 2 12 0 2 Solution EXAMPLE 10 For this equation, let = + u x 2. Then ( ) = + u x 2 , 2 2 and the original equation, ( ) ( ) + + + − = x x 2 11 2 12 0 2 becomes ( )( ) + − = + − = = − = u u u u u u 11 12 0 12 1 0 12 or 1 2 To solve for x, use = + u x 2 to obtain + =− + = =− =− x x x x 2 12 or 2 1 14 1 Check: ( ) ( ) = − − + + − + − x 14: 14 2 1114 2 12 2 ( ) ( ) = − + − − = − − = 12 11 12 12 144 132 12 0 2 ( ) ( ) =− −+ +−+−=+−= x 1: 12 1112 12 11112 0 2 The original equation has the solution set { } − − 14, 1. u x Let 2 . ( ) = + Factor. Solve. 8 Solve Absolute Value Equations On the real number line, the absolute value of a equals the distance from the origin to the point whose coordinate is a. For example, there are two points whose distance from the origin is 5 units, −5 and 5. So, the equation = x 5 will have the solution set { } −5, 5 . This leads to the following result: Equations Involving Absolute Value If a is a positive real number and if u is any algebraic expression, then = = = − u a u a u a is equivalent to or (5) Solution Solving an Equation Involving Absolute Value Solve the equation + = x 4 13. EXAMPLE 11 This follows the form of equation (5), where = + u x 4. There are two possibilities. + = = + = − = − x x x x 4 13 9 or 4 13 17 The solution set is { } −17, 9 . Now Work problem 51 9 Solve Equations by Factoring We have already solved certain quadratic equations by factoring. Let’s look at examples of other kinds of equations that can be solved by factoring.
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