3.5 Quadratic Equations

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3.5 Quadratic Equations

• OBJ To solve a quadratic equation by factoring

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DEF ? Standard form of a quadratic equation

• ax2 bx c 0
• NOTE ? Each equation contains a
• polynomial of the second degree.

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DEF ? Zero product property

• If mn 0, then m 0 or n 0 or both 0
• NOTE ? Solve some quadratic equations by
• Writing equation in standard form
• Factoring
• Setting each factor equal to 0

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EX ? 5 c 2 7c 6 0

• 3 5 2
• 2 1 3
• -3
• 3
• 2
• 10
• (5c 3)(c 2) 0
• c 3/5, -2

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EX ? 7t 20 3 t 2

• 3 t 2 7t 20 0
• 4 3 5
• 5 1 4
• -5
• 5
• 4
• 12
• (3t 5)(t 4) 0
• t 5/3, -4

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EX ? 36 25 x 2

• 25 x 2 36 0
• (5x 6)(5x 6) 0
• x 6/5

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EX ? 2 x 2 5x

• 2 x 2 5x 0
• x(2x 5) 0
• x 0, - 5/2

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EX? 7 n 2 14n 56 0

• 7 (n 2 2n 8) 0
• 7 (n 4)(n 2) 0
• n - 4, 2

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EX? y 4 5 y 2 4 0

• (y 2 4)(y 2 1) 0
• (y 2)(y 2)(y 1)(y 1) 0
• Y 2, 1

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EX? y 4 10 y 2 9 0

• (y 2 9)(y 2 1) 0
• (y 3)(y 3)(y 1)(y 1) 0
• Y 3, 1

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EX ? y 4 20 y 2

• y 4 y 2 20 0
• (y 2 5)(y 2 4) 0
• (y 2 5)(y 2)(y 2) 0
• Y iv 5, 2

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EX ? y 4 12 y 2

• y 4 y 2 12 0
• (y 2 4)(y 2 3) 0
• (y 2)(y 2)(y 2 3) 0
• Y 2, iv 3

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6.1 Square Roots

• OBJ ? To solve a quadratic equation by using
  the definition of square root
• DEF ? Square root
• If x 2 k, then x vk, for k 0

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EX ? 6 y 2 20 8 y 2

• 7y 2 28
• y 2 4
• y 2

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EX ? 3 n 2 9 7 n 2 35

• 44 4n 2
• 11 n 2
• v11 n

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7.3 The Quadratic Formula

• OBJ ? To solve a quadratic equation by using the
  quadratic formula
• DEF ? The quadratic formula
• x -b vb2 4ac
• 2a

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EX ? 4 x 2 7x 2 0

• x -(-7) v(-7)2 4(4)(2)
• 2(4)
• 7 v49 32
• 8
• 7 v17
• 8

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EX ? 9 x 2 12x 1

• 9 x 2 12x 1 0
• x -(-12)v(-12)2 4(9)(1)
• 2(9)
• 12 v144 36
• 18
• 12 v108
• 18
• 12 6v3
• 18

• 12 6v3
• 18
• 6(2 v3)
• 18
• 3
• 2 v3
• 3

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EX ? 6 x 2 5x 0

• x(6x 5) 0
• x 0, -5/6

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EX 8 ? 72 x 2 0

• x 2 72
• x 6v2

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8.3 Equations With Imaginary Number Solutions

• OBJ To solve an equation whose solutions are
  imaginary

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EX ? 2 x 2 7 6x

• 2 x 2 6x 7 0
• x -(-6)v(-6)2 4(2)(7)
• 2(2)
• 6 v36 56
• 4
• 6 v-20
• 4
• 6 2iv5
• 4

• 2(3 iv5)
• 4
• 2(3 iv5)
• 4
• 2
• 3 iv5
• 2

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EX ? 27 6 y 2 y 4

• y 4 6 y 2 27 0
• (y 2 9)(y 2 3) 0
• y 3i, v3