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**Exploring Quadratic Functions and Inequalities**

Advanced Algebra Chapter 6

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**Solving Quadratic Functions**

Solve the following equation. Square of a Binomial Solution:

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**Solving Quadratic Functions**

Multiply the following expressions. Is there a pattern? Shortcut Method ( x ) = x x 2×product of both terms 1st term last term square of 1st term square of last term

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**Solving Quadratic Functions**

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

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**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.

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**Solving Quadratic Functions by Completing the Square**

Step 3 Factor and Solve.

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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.

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**Quadratic Formula The Quadratic Formula**

The value of the discriminant, b2 – 4ac, determines the nature of the roots of a quadratic equation. The Discriminant

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**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.

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**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.

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**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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Homework

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