Chapter 10 Quadratic Equations and Functions

1. Chapter 10 Quadratic Equations and Functions Section 3 Solving Equations Quadratic in Form

2. Section 10.3 Objectives 1 Solve Equations That Are Quadratic in Form

3. Equations Quadratic in Form If a substitution u transforms an equation into one of the form au2 + bu + c = 0 then the original equation is called an equation quadratic in form. Solving Equations Quadratic in Form Step 1: Determine the appropriate substitution and write the equation in the form au2 + bu + c = 0. Step 2: Solve the equation au2 + bu + c = 0. Step 3: Solve for the variable in the original equation using the value of u found in Step 2. Step 4: Check. Verify your solutions.

4. Equations Quadratic in Form Example: Solve: x4 – 11x2 + 18 = 0 u2 – 11u + 18 = 0 Replace x4 with u2 and x2 with u. (u – 9)(u – 2) = 0 Factor. u – 9 = 0 or u – 2 = 0 Set each equation equal to zero. u = 9 u = 2 Solve for u. x2 = 9 x2 = 2 Replace u with x2. Solve for x. Continued.

5. Equations Quadratic in Form Example continued: x4 – 11x2 + 18 = 0 Check: (3)4 – 11(3)2 + 18 = 0 81 – 11(9) + 18 = 0 81 – 99 + 18 = 0   0 = 0 (–3)4 – 11(– 3)2 + 18 = 0 81 – 11(9) + 18 = 0 81 – 99 + 18 = 0   0 = 0

6. Equations Quadratic in Form Example: Solve: 2x1/2 – x1/4 – 1 = 0 2u2 – u – 1 = 0 Replace x1/2 with u2 and x1/4 with u. (2u + 1)(u – 1) = 0 Factor. 2u + 1 = 0 or u – 1 = 0 Set each equation equal to zero. Solve for u. Replace u with x1/4. Solve for x. Continued.

7. Check: is an extraneous root. Equations Quadratic in Form Example continued:  The solution set is {1}.