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2. Chapter 11 Quadratic Equations

3. 11.1 Review of Solving Equation by Factoring 11.2 The Square Root Property and Completing the Square 11.3 The Quadratic Formula Putting It All Together 11.4Equations in Quadratic Form 11.5Formulas and Applications 11 Quadratic Equations

4. Equations in Quadratic Form 11.4 In Chapters 8 and 10, we solved some equations that were not quadratic but could be rewritten in form of a quadratic equation,. Two such examples are: Rational equation (Ch.8) Rational equation (Ch.10) We will review how to solve each type of equation. Solve Quadratic Equations Resulting from Equations Containing Fractions or Radicals Solve. Example 1 Solution To solve an equation containing rational expressions, multiply the equation by the LCD of all the fractions to eliminate the denominators, then solve. Multiply both sides of the equation By the LCD of the fractions. Continued on next slide..

5. Multiply both sides of the equation By the LCD of the fractions. Distribute and divide out common factors. Distribute. Combine like terms. Write in the form Factor. Set each factor equal to zero. Solve. Recall that you must check the proposed solutions in the original Equation to be certain they do not make a denominator equal to Zero.

6. Solve. Example 2 Solution The first step in solving a radical equation is getting a radical on a side by itself. Subtract d from each side Square both sides. Write in the form Factor. Set each factor equal to zero.. Solve. Recall that you must check the proposed solutions in the original equation. 25 is an extraneous solution. The solution set is {16}.

7. Solve an Equation in Quadratic Form by Factoring Some equations that are not quadratic can be solved using the same methods that can be used to solve quadratic equations. These are called equations in quadratic form. Some examples of equations in quadratic form are listed in the table below. This pattern enables us to work with equations in quadratic form like we can work with quadratic equations.

8. Solve. Example 3 Solution The check is left to the student. The solution set is {-5, -2, 2, 5}. Factor. Set each factor equal to 0. Square root property

9. Solve. Example 4 Solution The check is left to the student. The solution set is {-64, 27}. Factor. Set each factor equal to 0. Isolate the constant. Cube both sides. Solve.

10. Solve an Equation in Quadratic Form Using Substitution Solve. Example 5 Solution Substitute Solve by factoring. Set each factor equal to 0. Solve for u. Be careful: u = 25 and u = 4 are not the solutions to. We still need to solve for x. To solve for x, substitute 25 for u and solve for x and then substitute 4 for u and solve for x. The check is left to the student. The solution set is {-5, -2, 2, 5}.

11. Solve. Example 6 Solution

13. Use Substitution for a Binomial to Solve a Quadratic Equation Solve. Example 7 Solution To binomial 4h + 1 appears as a squared quantity and as a linear quantity. Begin by using substitution. Substitute u for (4h+1). Factor. Set each factor equal to 0. Solve for u. Solve for a by substituting the values of u. Subtract 1. Multiply by ¼. Subtract 1. Divide by 4.