Chapter 8 Quadratic Functions

Chapter Sections 8.1 – Solving Quadratic Equations by Completing the Square 8.2 – Solving Quadratic Equations by the Quadratic Formulas 8.3 – Quadratic Equations: Applications and Problem Solving 8.4 – Writing Equations in Quadratic Form 8.5 – Graphing Quadratic Functions 8.6 – Quadratic and Other Inequalities in One Variable Chapter 1 Outline

Writing Equations in Quadratic Form § 8.4 Writing Equations in Quadratic Form

Solve Equations That Are Quadratic in Form Equations Quadratic in Form An equation that can be rewritten in the form au2 + bu + c = 0 for a ≠ 0, where u is an algebraic expression, is called quadratic in form.

Solve Equations That Are Quadratic in Form To Solve Equations Quadratic in Form Make a substitution that will result in an equation of the form au2 + bu + c = 0 for a ≠ 0 where u is a function of the original variable. Solve the equation au2 + bu + c = 0 for u. Replace u with the function of the original variable from step 1. Solve the resulting equation for the original variable. If, during step 4, you raise both sides of the equation to an even power, check for extraneous solutions by substituting the apparent solutions into the original equation.

Solve Equations That Are Quadratic in Form Example Solve x4 – 5x2 + 4 = 0. To obtain an equation quadratic in form, we will let u = x2. Then u2 = (x2)2 = x4. We now have a quadratic equation we can solve by factoring continued

Solve Equations That Are Quadratic in Form Next, we replace u with x2 and solve for x.

Solve Equations with Rational Exponents Whenever you raise both sides of an equation to an even power, you must check all apparent solutions in the original equation. Example Solve . We will let u=x1/4, then u2 = (x1/4)2 = x2/4 = x1/2 continued

Solve Equations with Rational Exponents Now substitute x1/4 for u and raise both sides of the equation to the fourth power and solve for x. continued

Solve Equations with Rational Exponents The two possible solutions are 81 and 16. However, since we raised both sides of an equation to an even power, we need to check for extraneous solutions. continued

Solve Equations with Rational Exponents Since 81 does not check, it is an extraneous solution. The only solution is 16.