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Exploring Quadratic Functions and Inequalities Advanced Algebra Chapter 6 Solving Quadratic Functions Solve the following equation. x 42 25 Solution: x 42 25 x 4 5 Square of a Binomial x 4 5 x45 x 9 x 1 Solving Quadratic Functions Multiply the following expressions. x 3x 3 x 2 6 x 9 2x 52x 5 4 x 2 20 x 25 Is there a pattern? 2×product of both terms Shortcut Method ( x + 6 )2 1st term last term = x2 + 12x + 36 square of 1st term square of last term Solving Quadratic Functions Try using the shortcut method with these. x 12 2 x2 2x 1 4 4 2 2 x x x 3 9 3 Now Try Backwards: x2 + 8x + 16 = ( x 4 )2 x2 – 4x + 4 = ( x 2 )2 x2 + x + ¼ = ( x + ½ )2 Solving Quadratic Functions by Completing the Square For example, solve the following equation by completing the square. x 2 3x 18 0 Step 1 Move the constant to the other side. x 2 3x ___ 18 Step 2 Notice the coefficient of the linear term is 3, or b = 3. Therefore, b 2 2 is the new constant needed to create a Square Binomial. Add this value to both sides. 2 3 3 2 x 3 x 18 2 2 2 Solving Quadratic Functions by Completing the Square Step 3 Factor and Solve. 2 3 3 x 2 3 x 18 2 2 2 x 3 9 2 2 x 2 x 9 3 2 2 9 3 x 2 2 2 6 x 3 2 3 72 9 x 2 4 4 3 81 x 2 4 3 9 x 2 2 3 9 2 2 12 x 6 2 Quadratic Formula Another way to solve quadratic equations is to use the quadratic formula. b b 4ac x 2a 2 This is derived from the standard form of the equation ax2 + bx + c = 0 by the process of completing the square. Quadratic Formula The Quadratic Formula b b 4ac x 2a 2 The Discriminant The value of the discriminant, b2 – 4ac, determines the nature of the roots of a quadratic equation. Discriminant b2 – 4ac b2 – 4ac Value Description b2 – 4ac = 0 b2 – 4ac < 0 b2 – 4ac > 0 is a perfect square Intersects the x-axis once. One real root. Does not intersect the x-axis. Two imaginary roots. Intersects the x-axis twice. Two real, irrational roots. Intersects the x-axis twice. Two real, rational roots. 10 y 10 y 5 Sample Graph -5 y 5 5 x x -5 x -5 -1 5 x -5 -1 5 -6 y 5 -5 Solving Quadratic Functions with the Quadratic Formula For example, solve the following equation with the quadratic formula. 4 x 2 25 20 x Step 1 Write quadratic equation in Standard Form. The discriminant, (–20)2 – 4(4)(25) = 0. There is one real, rational root. 4 x 20 x 25 0 2 Step 2 Substitute coefficients into quadratic formula. In this case a = 4, b = –20 and c = 25 x 20 20 24 2 4425 20 5 x 8 2 Solving Quadratic Functions with the Quadratic Formula For example, solve the following equation with the quadratic formula. 3x 2 2 5 x Step 1 Write quadratic equation in Standard Form. 3x 5 x 2 0 2 The discriminant, (–5)2 – 4(3)(2) = 1. There are two real, rational roots. Step 2 Substitute coefficients into quadratic formula. In this case a = 3, b = –5 and c = 2 x 5 5 23 2 432 6 x 1 6 4 2 x 6 3 Homework