Exploring Quadratic Functions and Inequalities Advanced Algebra Chapter 6

Solving Quadratic Functions Solve the following equation. Square of a Binomial Solution:

Solving Quadratic Functions Multiply the following expressions. Is there a pattern? Shortcut Method ( x + 6 )2 = x2 + 12x + 36 2×product of both terms 1st term last term square of 1st term square of last term

Solving Quadratic Functions Try using the shortcut method with these. Now Try Backwards: x2 + 8x + 16 = ( )2 x2 – 4x + 4 = ( )2 x2 + x + ¼ = ( )2 THINK!!! x + ½

Solving Quadratic Functions by Completing the Square For example, solve the following equation by completing the square. Step 1  Move the constant to the other side. Step 2  Notice the coefficient of the linear term is 3, or b = 3. Therefore, is the new constant needed to create a Square Binomial. Add this value to both sides.

Solving Quadratic Functions by Completing the Square Step 3  Factor and Solve.

Quadratic Formula Another way to solve quadratic equations is to use the quadratic formula. 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 The value of the discriminant, b2 – 4ac, determines the nature of the roots of a quadratic equation. The Discriminant

Discriminant  b2 – 4ac Value Description Sample Graph b2 – 4ac is a perfect square b2 – 4ac = 0 b2 – 4ac < 0 b2 – 4ac > 0 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.

Solving Quadratic Functions with the Quadratic Formula For example, solve the following equation with the quadratic formula. Step 1  Write quadratic equation in Standard Form. Step 2  Substitute coefficients into quadratic formula. In this case a = 4, b = –20 and c = 25 The discriminant, (–20)2 – 4(4)(25) = 0. There is one real, rational root.

Solving Quadratic Functions with the Quadratic Formula For example, solve the following equation with the quadratic formula. Step 1  Write quadratic equation in Standard Form. Step 2  Substitute coefficients into quadratic formula. In this case a = 3, b = –5 and c = 2 The discriminant, (–5)2 – 4(3)(2) = 1. There are two real, rational roots.

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