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**MTH 065 Elementary Algebra II**

Chapter 11 Quadratic Functions and Equations Section 11.1 Quadratic Equations

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**Geometric Representation of Completing the Square**

x x + 8 Area = x(x + 8)

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**Geometric Representation of Completing the Square**

x x2 8x x 8 Area = x2 + 8x

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**Geometric Representation of Completing the Square**

x x2 8x x 8 Area = x2 + 8x

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**Geometric Representation of Completing the Square**

x x2 8x x 4 4 Area = x2 + 8x

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**Geometric Representation of Completing the Square**

x x2 4x 4x x 4 4 Area = x2 + 8x

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**Geometric Representation of Completing the Square**

x x2 4x 4x x 4 4 Area = x2 + 8x

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**Geometric Representation of Completing the Square**

4 4x x x2 4x x 4 Area = x2 + 8x

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**Geometric Representation of Completing the Square**

4 4x ? x x2 4x x 4 Area = x2 + 8x + ?

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**Geometric Representation of Completing the Square**

4 4x 16 x x2 4x x 4 Area = x2 + 8x + 16

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**Geometric Representation of Completing the Square**

4 4x 16 x x2 4x x 4 Area = x2 + 8x + 16 = (x + 4)2

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**Terminology ax2 + bx + c = 0 f(x) = ax2 + bx + c Quadratic Equation**

Any equation equivalent to an equation with the form … ax2 + bx + c = 0 … where a, b, & c are constants and a ≠ 0. Quadratic Function Any function equivalent to the form … f(x) = ax2 + bx + c ... where a, b, & c are constants and a ≠ 0.

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**Review Results from Chapter 6**

Solve quadratic equations by graphing. Put into standard form: ax2 + bx + c = 0 Graph the function: f(x) = ax2 + bx + c Solutions are the x-intercepts. # of Solutions? , 1, or 2 Details of Graphs of Quadratic Functions – Section 11.6

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**Review Results from Chapter 6**

Solve quadratic equations by factoring. Put into standard form: ax2 + bx + c = 0 Factor the quadratic: (rx + m)(sx + n) = 0 Set each factor equal to zero and solve. # of Solutions? 0 does not factor (not factorable no solution) 1 factors as a perfect square (if it factors) 2 two different factors (if it factors)

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**Principle of Square Roots**

For any number k, if … … then …

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**Principle of Square Roots**

For any number k, if … … then … Why? Consider the following example … x2 = 9 x2 – 9 = 0 (x – 3)(x + 3) = 0 x = 3, –3

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**Application of the Principle of Square Roots**

Solve the equation … Note This example demonstrates how to solve a quadratic equation with no linear (bx) term.

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**Application of the Principle of Square Roots**

Solve the equation … Note Remember to always simplify radicals. no perfect squares no multiples of perfect squares no negatives

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**Application of the Principle of Square Roots**

Solve the equations …

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**Application of the Principle of Square Roots**

Solve the equation … But this does not factor …

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**Solving by “Completing the Square”**

Note: This polynomial does not factor.

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**Solving ax2 + bx + c = 0 by “Completing the Square”**

Basic Steps … Get into the form: ax2 + bx = d Divide through by a giving: x2 + mx = n Add the square of half of m to both sides. i.e. add Factor the left side (a perfect square). Solve using the Principle of Square Roots.

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