MTH 065 Elementary Algebra II Chapter 11 Quadratic Functions and Equations Section 11.1 Quadratic...
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Transcript of MTH 065 Elementary Algebra II Chapter 11 Quadratic Functions and Equations Section 11.1 Quadratic...
42510011 0010 1010 1101 0001 0100 1011
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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0011 0010 1010 1101 0001 0100 1011
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