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Quadratic Functions

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

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Quadratic Functions

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  1. Chapter 8 Quadratic Functions

  2. 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

  3. Solving Quadratic Equations by Completing the Square § 8.1

  4. Quadratic Equations A quadratic equation is an equation of the form ax2 + bx + c = 0 where a, b, and c are real numbers and a 0. In Section 5.8 we solved quadratic equations by factoring. In this section we introduce two additional procedures used to solve quadratic equations: the square root property and completing the square.

  5. Square Root Property Square Root Property If x2 = a, where a is a real number, then x = ± √a. Solve the equation x2 = 49.

  6. Square Root Property Solve the equation x2 - 9 = 0. Solve the equation x2 + 10 = 85.

  7. Understand Perfect Square Trinomials A perfect square trinomialis a trinomial that can be expressed as the square of a binomial. x2 – 10x + 25 = (x – 5) (x – 5) = (x – 5)2 a2 + 8a + 16 = (a + 4) (a + 4) = (a + 4)2 p2 – 14p + 49 = (p – 7) (p – 7) = (p – 7)2 Note that in every perfect square trinomial,the constant term is the square of one-half the coefficient of the x-term.

  8. Completing the Square To Solve a Quadratic Equation by Completing the Square • Use the multiplication (or division) property of equality, if necessary, to make the leading coefficient 1. • Rewrite the equation with the constant by itself on the right side of the equation. • Take one-half the numerical coefficient of the first-degree term, square it, and add this quantity to both sides of the equation. • Factor the trinomial as the square of a binomial. • Use the square root property to take the square root of both sides of the equation. • Solve for the variable. • Check your solutions in the original equation.

  9. Completing the Square Example Solve the equation x2 + 6x + 5 = 0 by completing the square. Step 1Since the leading coefficient is 1, step 1 is not necessary. Step 2Subtract 5 from both sides of the equation. continued

  10. Completing the Square Step 3Determine the square of one-half the numerical coefficient of the first degree term, 6. Add this value to both sides of the equation. continued

  11. Completing the Square Step 4By following this procedure, we produce a perfect square trinomial on the left side of the equation. The expression x2 + 6x + 9 is a perfect square trinomial that can be factored as (x + 3)2. Step 5 Use the square root property. continued

  12. Completing the Square Step 6Finally, solve for x by subtracting 3 from both sides of the equation. Step 7 Check both solutions in the original equation.

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