SECTION A.6 Solving Equations A53 7 Solve Equations Quadratic in Form The equation + − = x x 12 0 4 2 is not quadratic in x, but it is quadratic in x .2 That is, letting = u x2 results in + − = u u 12 0, 2 a quadratic equation. This equation can be solved for u and, in turn, = u x2 can be solved for x to find the solutions of the original equation. In general, if an appropriate substitution u transforms an equation into one of the form + + = ≠ au bu c a 0 0 2 then the original equation is called an equation of the quadratic type or an equation quadratic in form. The difficulty of solving such an equation lies in the determination that the equation is, in fact, quadratic in form. After you are told an equation is quadratic in form, it is easy enough to see it, but some practice is needed to enable you to recognize such equations on your own. Solving a Quadratic Equation by Using the Quadratic Formula Find the real solutions, if any, of the equation + = x x 3 2 4 2 Solution EXAMPLE 9 The equation, as given, is not in standard form. + = − + = + + = x x x x ax bx c 3 2 4 3 4 2 0 0 2 2 2 With = = − a b 3, 4, and = c 2, the discriminant is ( ) − =− −⋅⋅= − =− b ac 4 4 4 3 2 16 24 8 2 2 Since − < b ac 4 0, 2 the equation has no real solution. Subtract x4 from both sides to put the equation in standard form. Compare to standard form. Now Work problem 119 SUMMARY Procedure for Solving a Quadratic Equation To solve a quadratic equation, first put it in standard form: + + = ax bx c 0 2 Then: Step 1 Identify a b , , and c. Step 2 Evaluate the discriminant, − b ac 4 . 2 Step 3 • If the discriminant is negative, the equation has no real solution. • If the discriminant is zero, the equation has one real solution, a double root. • If the discriminant is positive, the equation has two distinct real solutions. If you can easily spot factors, use the factoring method to solve the equation. Otherwise, use the quadratic formula or the method of completing the square.
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