Algebra III Section 4'2

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Algebra IIISection 4.2

• Standard
• Students will be able to solve polynomial
  functions.
• Essential Question
• How do you solve a quadratic equation?

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Terms

• Quadratic Equation Standard Form
• Ax2 Bx C 0
• Quadratic Formula
• Factoring
• Completing the Square

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Quadratic EquationsStandard Form

• ax2 bx c 0
• 2x2 3x 4 0
• x2 49 0
• 5x2 25 0

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Discriminant

• b2 4ac
• ax2 bx c 0
• b2 4ac gt 0, two real roots
• b2 4ac 0, one real root
• b2 4ac lt 0, two imaginary roots

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Describe the nature of the roots

• 6x2 7x 3 0
• two real
• x2 5x 9 0
• two imaginary
• 36x2 84x 49 0
• one real root

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Solving Quadratic Equations

• Graphing
• Factoring
• Quadratic formula
• Completing the Square

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Solve by Graphing

• Find the zeroes (yesterday)
• x2 6x 16 0
• Graph
• (CALC), 2zero
• Roots -2, 8

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Solving by Graphing

• Positive
• Very easy with the GC
• Negative
• Will not give you imaginary roots

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Solving by Factoring

• Factor trinomial into two binomials
• x2 6x 5 ? (x 5)(x 1)
• x2 4x 4 ? (x 2)(x 2)
• x2 5x 6 ? (x 1)(x 6)
• x2 25 ? (x 5)(x 5)
• Set each binomial equal to zero and solve for x
• (x 5)(x 1) 0
• x -5, x -1

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Solving by Factoring

• Positive
• Easy to use
• Negative
• Hard to use if coefficient of x2 ? 1
• Hard to find Imaginary roots

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Quadratic Formula

• -b vb2 4ac
• 2a
• Use ax2 bx c 0
• 5x2 3x 6 0
• a 5, b 3, c -6

x
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Quadratic Formula

• 5x2 3x 6
• a 5, b 3, c -6
• -3 v32 4(5)(-6)
• 2(5)
• -3 v129
• 10

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Quadratic Formula

• 2x2 5x 4 0
• x -5 iv7
• 4
• 6x2 x 2 0
• x -?, ½

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Quadratic Formula

• Positive
• Always gives all roots
• Negative
• Not always easy to use

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Solving by Completing the square

• Use the form x2 bx c 0
• May have to divide by a (leading coefficient)
• Move c to right side of equation (x2 bx c)
• Divide b by 2, then square the result (b/2)2
• Add the result to both sides x2 bx (b/2)2 c
  (b/2)2
• Factor the left side (should be a perfect square
  trinomial) (x b/2)2 c (b/2)2
• Take the square root of both sides
• Solve for x, simplify

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Solve by Completing the Square

• x2 4x 6 0
• x2 4x 6
• x2 4x 4 6 4
• (x 2)2 10
• x 2 v10
• x -2 v10

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Solve by completing the Square

• 2x2 4x 10 0
• x2 2x 5 0
• x2 2x 5
• x2 2x 1 5 1
• (x 1)2 6
• x 1 v6
• x 1 v6

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Completing the Square

• Positives
• Will give all roots
• Negative
• Harder if b is an odd number

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Review
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Homework

• Page 219
• 12 - 17, 20 25
• Use any method you choose