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MTH 065Elementary Algebra II

Chapter 11

Quadratic Functions and Equations

Section 11.1

Quadratic Equations

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Geometric Representation ofCompleting the Square

x

x + 8Area = x(x + 8)

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Geometric Representation ofCompleting the Square

x

x 8

x2 8x

Area = x2 + 8x

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Geometric Representation ofCompleting the Square

x

x 8

x2 8x

Area = x2 + 8x

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Geometric Representation ofCompleting the Square

x

x 4

x2 8x

4Area = x2 + 8x

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Geometric Representation ofCompleting the Square

x

x 4

x2 4x

4

4x

Area = x2 + 8x

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Geometric Representation ofCompleting the Square

x

x 4

x2 4x

4

4x

Area = x2 + 8x

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Geometric Representation ofCompleting the Square

x

x 4

x2 4x

4 4x

Area = x2 + 8x

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Geometric Representation ofCompleting the Square

x

x 4

x2 4x

4 4x ?

Area = x2 + 8x + ?

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Geometric Representation ofCompleting the Square

x

x 4

x2 4x

4 4x 16

Area = x2 + 8x + 16

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Geometric Representation ofCompleting the Square

x

x 4

x2 4x

4 4x 16

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

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Terminology

• Quadratic EquationAny equation equivalent to an equation with the form …

ax2 + bx + c = 0… where a, b, & c are constants and a ≠ 0.

• Quadratic FunctionAny 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? 0, 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 …

2x k

, x k k

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Principle of Square Roots

For any number k, if …

… then …2x k x k

Why? Consider the following example …

x2 = 9 x2 – 9 = 0 (x – 3)(x + 3) = 0 x = 3, –3

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Application of thePrinciple of Square Roots

Solve the equation …

3x

2 3x

25 15x

25 15 0x NoteThis example demonstrates how to solve a quadratic equation with no linear (bx) term.

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Application of thePrinciple of Square Roots

Solve the equation …

2 3x

25 15x

25 15 0x

3 3x i

Note

Remember to always simplify radicals.

• no perfect squares• no multiples of perfect

squares• no negatives

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Application of thePrinciple of Square Roots

Solve the equations …

3 2x

3 2x

2( 3) 4x

5, 1x

5 7x

5 7x

2( 5) 7x

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Application of thePrinciple of Square Roots

Solve the equation …

2 8 5 0x x

2 8 16 11x x

But this does not factor …

2( 4) 11x

4 11x

4 11x

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Solving by“Completing the Square”

2 6 7 0x x Note: This polynomial does not factor.

2 6 7x x 22 36 7 9x x 2( 23)x

3 2x

3 2x

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

• Basic Steps …

1. Get into the form: ax2 + bx = d

2. Divide through by a giving: x2 + mx = n

3. Add the square of half of m to both sides.

• i.e. add

4. Factor the left side (a perfect square).

5. Solve using the Principle of Square Roots.

2

2

m