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6.5 – Solving Equations with Quadratic Techniques

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Quadratic equations are in the form: ax 2 + bx + c,

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Quadratic equations are in the form: ax 2 + bx + c, where a, b, & c are integers

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Quadratic equations are in the form: ax 2 + bx + c, where a, b, & c are integers exs.

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Quadratic equations are in the form: ax 2 + bx + c, where a, b, & c are integers exs. x 2 + 5x + 2

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Quadratic equations are in the form: ax 2 + bx + c, where a, b, & c are integers exs. x 2 + 5x + 2 2x 2 – 18x + 13

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Quadratic equations are in the form: ax 2 + bx + c, where a, b, & c are integers exs. x 2 + 5x + 2 2x 2 – 18x + 13 x 2 – 9

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Quadratic equations are in the form: ax 2 + bx + c, where a, b, & c are integers exs. x 2 + 5x + 2 2x 2 – 18x + 13 x 2 – 9 x 2 + 0x – 9

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Quadratic equations are in the form: ax 2 + bx + c, where a, b, & c are integers exs. x 2 + 5x + 2 2x 2 – 18x + 13 x 2 – 9 x 2 + 0x – 9 2x 2 + 8x

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Quadratic equations are in the form: ax 2 + bx + c, where a, b, & c are integers exs. x 2 + 5x + 2 2x 2 – 18x + 13 x 2 – 9 x 2 + 0x – 9 2x 2 + 8x 2x 2 + 8x + 0

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Quadratic equations are in the form: ax 2 + bx + c, where a, b, & c are integers exs. x 2 + 5x + 2 2x 2 – 18x + 13 x 2 – 9 x 2 + 0x – 9 2x 2 + 8x 2x 2 + 8x + 0 NOTE: Must have the “x 2 ” term to be a quadratic!

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Ex. 1 Write each expression in quadratic form, if possible.

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a. x 4 + 13x 2 + 36

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Ex. 1 Write each expression in quadratic form, if possible. a. x 4 + 13x 2 + 36 (x 2 ) 2

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Ex. 1 Write each expression in quadratic form, if possible. a. x 4 + 13x 2 + 36 (x 2 ) 2 + 13(x 2 )

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Ex. 1 Write each expression in quadratic form, if possible. a. x 4 + 13x 2 + 36 (x 2 ) 2 + 13(x 2 ) + 36

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Ex. 1 Write each expression in quadratic form, if possible. a. x 4 + 13x 2 + 36 (x 2 ) 2 + 13(x 2 ) + 36

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Ex. 1 Write each expression in quadratic form, if possible. a. x 4 + 13x 2 + 36 (x 2 ) 2 + 13(x 2 ) + 36 b. 16x 6 – 625

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Ex. 1 Write each expression in quadratic form, if possible. a. x 4 + 13x 2 + 36 (x 2 ) 2 + 13(x 2 ) + 36 b. 16x 6 – 625 (4x 3 ) 2

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Ex. 1 Write each expression in quadratic form, if possible. a. x 4 + 13x 2 + 36 (x 2 ) 2 + 13(x 2 ) + 36 b. 16x 6 – 625 (4x 3 ) 2 – 625

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Ex. 1 Write each expression in quadratic form, if possible. a. x 4 + 13x 2 + 36 (x 2 ) 2 + 13(x 2 ) + 36 b. 16x 6 – 625 (4x 3 ) 2 – 625 c. x ½ – 9x ¼ + 16

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Ex. 1 Write each expression in quadratic form, if possible. a. x 4 + 13x 2 + 36 (x 2 ) 2 + 13(x 2 ) + 36 b. 16x 6 – 625 (4x 3 ) 2 – 625 c. x ½ – 9x ¼ + 16 (x ¼ ) 2

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Ex. 1 Write each expression in quadratic form, if possible. a. x 4 + 13x 2 + 36 (x 2 ) 2 + 13(x 2 ) + 36 b. 16x 6 – 625 (4x 3 ) 2 – 625 c. x ½ – 9x ¼ + 16 (x ¼ ) 2 – 9(x ¼ )

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Ex. 1 Write each expression in quadratic form, if possible. a. x 4 + 13x 2 + 36 (x 2 ) 2 + 13(x 2 ) + 36 b. 16x 6 – 625 (4x 3 ) 2 – 625 c. x ½ – 9x ¼ + 16 (x ¼ ) 2 – 9(x ¼ ) + 16

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Ex. 1 Write each expression in quadratic form, if possible. a. x 4 + 13x 2 + 36 (x 2 ) 2 + 13(x 2 ) + 36 b. 16x 6 – 625 (4x 3 ) 2 – 625 c. x ½ – 9x ¼ + 16 (x ¼ ) 2 – 9(x ¼ ) + 16

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Ex. 2 Solve each equation. a. x 4 = 16

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Ex. 2 Solve each equation. a. x 4 = 16 -16 -16

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Ex. 2 Solve each equation. a. x 4 = 16 -16 -16 x 4 – 16 = 0

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Ex. 2 Solve each equation. a. x 4 = 16 -16 -16 x 4 – 16 = 0 (x 2 ) 2 – 16 = 0

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Ex. 2 Solve each equation. a. x 4 = 16 -16 -16 x 4 – 16 = 0 (x 2 ) 2 – 16 = 0

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Ex. 2 Solve each equation. a. x 4 = 16 -16 -16 x 4 – 16 = 0 (x 2 ) 2 – 16 = 0 ()() = 0

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Ex. 2 Solve each equation. a. x 4 = 16 -16 -16 x 4 – 16 = 0 (x 2 ) 2 – 16 = 0 ()() = 0 (x 2 )(x 2 ) = 0

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Ex. 2 Solve each equation. a. x 4 = 16 -16 -16 x 4 – 16 = 0 (x 2 ) 2 – 16 = 0 ()() = 0 (x 2 – 4)(x 2 + 4) = 0

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Ex. 2 Solve each equation. a. x 4 = 16 -16 -16 x 4 – 16 = 0 (x 2 ) 2 – 16 = 0 ()() = 0 (x 2 – 4)(x 2 + 4) = 0 x 2 – 4 = 0

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Ex. 2 Solve each equation. a. x 4 = 16 -16 -16 x 4 – 16 = 0 (x 2 ) 2 – 16 = 0 ()() = 0 (x 2 – 4)(x 2 + 4) = 0 x 2 – 4 = 0 OR x 2 + 4 = 0

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Ex. 2 Solve each equation. a. x 4 = 16 -16 -16 x 4 – 16 = 0 (x 2 ) 2 – 16 = 0 ()() = 0 (x 2 – 4)(x 2 + 4) = 0 x 2 – 4 = 0 OR x 2 + 4 = 0 ( )( ) = 0

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Ex. 2 Solve each equation. a. x 4 = 16 -16 -16 x 4 – 16 = 0 (x 2 ) 2 – 16 = 0 ()() = 0 (x 2 – 4)(x 2 + 4) = 0 x 2 – 4 = 0 OR x 2 + 4 = 0 ( )( ) = 0 (x )(x ) = 0

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Ex. 2 Solve each equation. a. x 4 = 16 -16 -16 x 4 – 16 = 0 (x 2 ) 2 – 16 = 0 ()() = 0 (x 2 – 4)(x 2 + 4) = 0 x 2 – 4 = 0 OR x 2 + 4 = 0 ( )( ) = 0 (x – 2)(x + 2) = 0

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Ex. 2 Solve each equation. a. x 4 = 16 -16 -16 x 4 – 16 = 0 (x 2 ) 2 – 16 = 0 ()() = 0 (x 2 – 4)(x 2 + 4) = 0 x 2 – 4 = 0 OR x 2 + 4 = 0 ( )( ) = 0 (x – 2)(x + 2) = 0 x – 2 = 0 OR x + 2 = 0

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Ex. 2 Solve each equation. a. x 4 = 16 -16 -16 x 4 – 16 = 0 (x 2 ) 2 – 16 = 0 ()() = 0 (x 2 – 4)(x 2 + 4) = 0 x 2 – 4 = 0 OR x 2 + 4 = 0 ( )( ) = 0 (x – 2)(x + 2) = 0 x – 2 = 0 OR x + 2 = 0 +2 +2 -2 -2

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Ex. 2 Solve each equation. a. x 4 = 16 -16 -16 x 4 – 16 = 0 (x 2 ) 2 – 16 = 0 ()() = 0 (x 2 – 4)(x 2 + 4) = 0 x 2 – 4 = 0 OR x 2 + 4 = 0 ( )( ) = 0 (x – 2)(x + 2) = 0 x – 2 = 0 OR x + 2 = 0 +2 +2 -2 -2 x = 2 OR x = -2

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Ex. 2 Solve each equation. a. x 4 = 16 -16 -16 x 4 – 16 = 0 (x 2 ) 2 – 16 = 0 ()() = 0 (x 2 – 4)(x 2 + 4) = 0 x 2 – 4 = 0 OR x 2 + 4 = 0 ( )( ) = 0 (x – 2)(x + 2) = 0 x – 2 = 0 OR x + 2 = 0 +2 +2 -2 -2 x = 2 OR x = -2

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Ex. 2 Solve each equation. a. x 4 = 16 -16 -16 x 4 – 16 = 0 (x 2 ) 2 – 16 = 0 ()() = 0 (x 2 – 4)(x 2 + 4) = 0 x 2 – 4 = 0 OR x 2 + 4 = 0 ( )( ) = 0 OR ( )( ) = 0 (x – 2)(x + 2) = 0 x – 2 = 0 OR x + 2 = 0 +2 +2 -2 -2 x = 2 OR x = -2

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Ex. 2 Solve each equation. a. x 4 = 16 -16 -16 x 4 – 16 = 0 (x 2 ) 2 – 16 = 0 ()() = 0 (x 2 – 4)(x 2 + 4) = 0 x 2 – 4 = 0 OR x 2 + 4 = 0 ( )( ) = 0 OR ( )( ) = 0 (x – 2)(x + 2) = 0 OR x 2 + 4 = 0 x – 2 = 0 OR x + 2 = 0 +2 +2 -2 -2 x = 2 OR x = -2

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Ex. 2 Solve each equation. a. x 4 = 16 -16 -16 x 4 – 16 = 0 (x 2 ) 2 – 16 = 0 ()() = 0 (x 2 – 4)(x 2 + 4) = 0 x 2 – 4 = 0 OR x 2 + 4 = 0 ( )( ) = 0 OR ( )( ) = 0 (x – 2)(x + 2) = 0 OR x 2 + 4 = 0 x – 2 = 0 OR x + 2 = 0 OR x 2 = -4 +2 +2 -2 -2 x = 2 OR x = -2

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Ex. 2 Solve each equation. a. x 4 = 16 -16 -16 x 4 – 16 = 0 (x 2 ) 2 – 16 = 0 ()() = 0 (x 2 – 4)(x 2 + 4) = 0 x 2 – 4 = 0 OR x 2 + 4 = 0 ( )( ) = 0 OR ( )( ) = 0 (x – 2)(x + 2) = 0 OR x 2 + 4 = 0 x – 2 = 0 OR x + 2 = 0 OR x 2 = -4 +2 +2 -2 -2 √x 2 = √-4 x = 2 OR x = -2

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Ex. 2 Solve each equation. a. x 4 = 16 -16 -16 x 4 – 16 = 0 (x 2 ) 2 – 16 = 0 ()() = 0 (x 2 – 4)(x 2 + 4) = 0 x 2 – 4 = 0 OR x 2 + 4 = 0 ( )( ) = 0 OR ( )( ) = 0 (x – 2)(x + 2) = 0 OR x 2 + 4 = 0 x – 2 = 0 OR x + 2 = 0 OR x 2 = -4 +2 +2 -2 -2 √x 2 = √-4 x = 2 OR x = -2 OR x = ±2i

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Ex. 2 Solve each equation. a. x 4 = 16 -16 -16 x 4 – 16 = 0 (x 2 ) 2 – 16 = 0 ()() = 0 (x 2 – 4)(x 2 + 4) = 0 x 2 – 4 = 0 OR x 2 + 4 = 0 ( )( ) = 0 OR ( )( ) = 0 (x – 2)(x + 2) = 0 OR x 2 + 4 = 0 x – 2 = 0 OR x + 2 = 0 OR x 2 = -4 +2 +2 -2 -2 √x 2 = √-4 x = 2 OR x = -2 OR x = ±2i

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b. x 4 + 11x 2 + 18 = 0

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(x 2 ) 2 + 11(x 2 ) + 18 = 0

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b. x 4 + 11x 2 + 18 = 0 (x 2 ) 2 + 11(x 2 ) + 18 = 0 ()() = 0

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b. x 4 + 11x 2 + 18 = 0 (x 2 ) 2 + 11(x 2 ) + 18 = 0 ()() = 0 (x 2 )(x 2 ) = 0

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b. x 4 + 11x 2 + 18 = 0 (x 2 ) 2 + 11(x 2 ) + 18 = 0 ()() = 0 (x 2 + 9)(x 2 + 2) = 0

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b. x 4 + 11x 2 + 18 = 0 (x 2 ) 2 + 11(x 2 ) + 18 = 0 ()() = 0 (x 2 + 9)(x 2 + 2) = 0 x 2 + 9 = 0

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b. x 4 + 11x 2 + 18 = 0 (x 2 ) 2 + 11(x 2 ) + 18 = 0 ()() = 0 (x 2 + 9)(x 2 + 2) = 0 x 2 + 9 = 0 OR x 2 + 2 = 0

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b. x 4 + 11x 2 + 18 = 0 (x 2 ) 2 + 11(x 2 ) + 18 = 0 ()() = 0 (x 2 + 9)(x 2 + 2) = 0 x 2 + 9 = 0 OR x 2 + 2 = 0

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b. x 4 + 11x 2 + 18 = 0 (x 2 ) 2 + 11(x 2 ) + 18 = 0 ()() = 0 (x 2 + 9)(x 2 + 2) = 0 x 2 + 9 = 0 OR x 2 + 2 = 0 ( )( ) = 0

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b. x 4 + 11x 2 + 18 = 0 (x 2 ) 2 + 11(x 2 ) + 18 = 0 ()() = 0 (x 2 + 9)(x 2 + 2) = 0 x 2 + 9 = 0 OR x 2 + 2 = 0 (x + )(x + ) = 0

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b. x 4 + 11x 2 + 18 = 0 (x 2 ) 2 + 11(x 2 ) + 18 = 0 ()() = 0 (x 2 + 9)(x 2 + 2) = 0 x 2 + 9 = 0 OR x 2 + 2 = 0 ( )( ) = 0 x 2 + 9 = 0

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b. x 4 + 11x 2 + 18 = 0 (x 2 ) 2 + 11(x 2 ) + 18 = 0 ()() = 0 (x 2 + 9)(x 2 + 2) = 0 x 2 + 9 = 0 OR x 2 + 2 = 0 ( )( ) = 0 ()() = 0 x 2 + 9 = 0 -9 -9

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b. x 4 + 11x 2 + 18 = 0 (x 2 ) 2 + 11(x 2 ) + 18 = 0 ()() = 0 (x 2 + 9)(x 2 + 2) = 0 x 2 + 9 = 0 OR x 2 + 2 = 0 ( )( ) = 0 ()() = 0 x 2 + 9 = 0 -9 -9 x 2 = -9

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b. x 4 + 11x 2 + 18 = 0 (x 2 ) 2 + 11(x 2 ) + 18 = 0 ()() = 0 (x 2 + 9)(x 2 + 2) = 0 x 2 + 9 = 0 OR x 2 + 2 = 0 ( )( ) = 0 ()() = 0 x 2 + 9 = 0 -9 -9 x 2 = -9 √x 2 = √-9

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b. x 4 + 11x 2 + 18 = 0 (x 2 ) 2 + 11(x 2 ) + 18 = 0 ()() = 0 (x 2 + 9)(x 2 + 2) = 0 x 2 + 9 = 0 OR x 2 + 2 = 0 ( )( ) = 0 ()() = 0 x 2 + 9 = 0 -9 -9 x 2 = -9 √x 2 = √-9 x = ±3i

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b. x 4 + 11x 2 + 18 = 0 (x 2 ) 2 + 11(x 2 ) + 18 = 0 ()() = 0 (x 2 + 9)(x 2 + 2) = 0 x 2 + 9 = 0 OR x 2 + 2 = 0 ( )( ) = 0 ()() = 0 x 2 + 9 = 0 -9 -9 x 2 = -9 √x 2 = √-9 x = ±3i

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b. x 4 + 11x 2 + 18 = 0 (x 2 ) 2 + 11(x 2 ) + 18 = 0 ()() = 0 (x 2 + 9)(x 2 + 2) = 0 x 2 + 9 = 0 OR x 2 + 2 = 0 ( )( ) = 0 (x +)(x + ) = 0 x 2 + 9 = 0 -9 -9 x 2 = -9 √x 2 = √-9 x = ±3i

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b. x 4 + 11x 2 + 18 = 0 (x 2 ) 2 + 11(x 2 ) + 18 = 0 ()() = 0 (x 2 + 9)(x 2 + 2) = 0 x 2 + 9 = 0 OR x 2 + 2 = 0 ( )( ) = 0 ()() = 0 x 2 + 9 = 0 OR x 2 + 2 = 0 -9 -9 x 2 = -9 √x 2 = √-9 x = ±3i

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b. x 4 + 11x 2 + 18 = 0 (x 2 ) 2 + 11(x 2 ) + 18 = 0 ()() = 0 (x 2 + 9)(x 2 + 2) = 0 x 2 + 9 = 0 OR x 2 + 2 = 0 ( )( ) = 0 ()() = 0 x 2 + 9 = 0 OR x 2 + 2 = 0 -9 -9 -2 -2 x 2 = -9 √x 2 = √-9 x = ±3i

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b. x 4 + 11x 2 + 18 = 0 (x 2 ) 2 + 11(x 2 ) + 18 = 0 ()() = 0 (x 2 + 9)(x 2 + 2) = 0 x 2 + 9 = 0 OR x 2 + 2 = 0 ( )( ) = 0 ()() = 0 x 2 + 9 = 0 OR x 2 + 2 = 0 -9 -9 -2 -2 x 2 = -9 OR x 2 = -2 √x 2 = √-9 x = ±3i

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b. x 4 + 11x 2 + 18 = 0 (x 2 ) 2 + 11(x 2 ) + 18 = 0 ()() = 0 (x 2 + 9)(x 2 + 2) = 0 x 2 + 9 = 0 OR x 2 + 2 = 0 ( )( ) = 0 ()() = 0 x 2 + 9 = 0 OR x 2 + 2 = 0 -9 -9 -2 -2 x 2 = -9 OR x 2 = -2 √x 2 = √-9 OR √x 2 = √-2 x = ±3i

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b. x 4 + 11x 2 + 18 = 0 (x 2 ) 2 + 11(x 2 ) + 18 = 0 ()() = 0 (x 2 + 9)(x 2 + 2) = 0 x 2 + 9 = 0 OR x 2 + 2 = 0 ( )( ) = 0 ()() = 0 x 2 + 9 = 0 OR x 2 + 2 = 0 -9 -9 -2 -2 x 2 = -9 OR x 2 = -2 √x 2 = √-9 OR √x 2 = √-2 x = ±3i OR x = ±i√2

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b. x 4 + 11x 2 + 18 = 0 (x 2 ) 2 + 11(x 2 ) + 18 = 0 ()() = 0 (x 2 + 9)(x 2 + 2) = 0 x 2 + 9 = 0 OR x 2 + 2 = 0 ( )( ) = 0 ()() = 0 x 2 + 9 = 0 OR x 2 + 2 = 0 -9 -9 -2 -2 x 2 = -9 OR x 2 = -2 √x 2 = √-9 OR √x 2 = √-2 x = ±3i OR x = ±i√2

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c. x ½ – 6x ¼ – 16 = 0

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(x ¼ ) 2 – 6(x ¼ ) – 16 = 0

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c. x ½ – 6x ¼ – 16 = 0 (x ¼ ) 2 – 6(x ¼ ) – 16 = 0 ()() = 0

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c. x ½ – 6x ¼ – 16 = 0 (x ¼ ) 2 – 6(x ¼ ) – 16 = 0 ()() = 0 (x ¼ )(x ¼ ) = 0

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c. x ½ – 6x ¼ – 16 = 0 (x ¼ ) 2 – 6(x ¼ ) – 16 = 0 ()() = 0 (x ¼ – 8)(x ¼ + 2) = 0

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c. x ½ – 6x ¼ – 16 = 0 (x ¼ ) 2 – 6(x ¼ ) – 16 = 0 ()() = 0 (x ¼ – 8)(x ¼ + 2) = 0 x ¼ – 8 = 0

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c. x ½ – 6x ¼ – 16 = 0 (x ¼ ) 2 – 6(x ¼ ) – 16 = 0 ()() = 0 (x ¼ – 8)(x ¼ + 2) = 0 x ¼ – 8 = 0 OR x ¼ + 2 = 0

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c. x ½ – 6x ¼ – 16 = 0 (x ¼ ) 2 – 6(x ¼ ) – 16 = 0 ()() = 0 (x ¼ – 8)(x ¼ + 2) = 0 x ¼ – 8 = 0 OR x ¼ + 2 = 0 ( )( ) = 0

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c. x ½ – 6x ¼ – 16 = 0 (x ¼ ) 2 – 6(x ¼ ) – 16 = 0 ()() = 0 (x ¼ – 8)(x ¼ + 2) = 0 x ¼ – 8 = 0 OR x ¼ + 2 = 0 ( )( ) = 0 x ¼ – 8 = 0

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c. x ½ – 6x ¼ – 16 = 0 (x ¼ ) 2 – 6(x ¼ ) – 16 = 0 ()() = 0 (x ¼ – 8)(x ¼ + 2) = 0 x ¼ – 8 = 0 OR x ¼ + 2 = 0 ( )( ) = 0 x ¼ – 8 = 0 +8 +8

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c. x ½ – 6x ¼ – 16 = 0 (x ¼ ) 2 – 6(x ¼ ) – 16 = 0 ()() = 0 (x ¼ – 8)(x ¼ + 2) = 0 x ¼ – 8 = 0 OR x ¼ + 2 = 0 ( )( ) = 0 x ¼ – 8 = 0 +8 +8 x ¼ = 8

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c. x ½ – 6x ¼ – 16 = 0 (x ¼ ) 2 – 6(x ¼ ) – 16 = 0 ()() = 0 (x ¼ – 8)(x ¼ + 2) = 0 x ¼ – 8 = 0 OR x ¼ + 2 = 0 ( )( ) = 0 x ¼ – 8 = 0 +8 +8 x ¼ = 8 (x ¼ ) 4 = (8) 4

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c. x ½ – 6x ¼ – 16 = 0 (x ¼ ) 2 – 6(x ¼ ) – 16 = 0 ()() = 0 (x ¼ – 8)(x ¼ + 2) = 0 x ¼ – 8 = 0 OR x ¼ + 2 = 0 ( )( ) = 0 x ¼ – 8 = 0 +8 +8 x ¼ = 8 (x ¼ ) 4 = (8) 4 x = 4096

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c. x ½ – 6x ¼ – 16 = 0 (x ¼ ) 2 – 6(x ¼ ) – 16 = 0 ()() = 0 (x ¼ – 8)(x ¼ + 2) = 0 x ¼ – 8 = 0 OR x ¼ + 2 = 0 ( )( ) = 0 x ¼ – 8 = 0 +8 +8 x ¼ = 8 (x ¼ ) 4 = (8) 4 x = 4096

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c. x ½ – 6x ¼ – 16 = 0 (x ¼ ) 2 – 6(x ¼ ) – 16 = 0 ()() = 0 (x ¼ – 8)(x ¼ + 2) = 0 x ¼ – 8 = 0 OR x ¼ + 2 = 0 ( )( ) = 0 ()( ) = 0 x ¼ – 8 = 0 +8 +8 x ¼ = 8 (x ¼ ) 4 = (8) 4 x = 4096

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c. x ½ – 6x ¼ – 16 = 0 (x ¼ ) 2 – 6(x ¼ ) – 16 = 0 ()() = 0 (x ¼ – 8)(x ¼ + 2) = 0 x ¼ – 8 = 0 OR x ¼ + 2 = 0 ( )( ) = 0 ()( ) = 0 x ¼ – 8 = 0 OR x ¼ + 2 = 0 +8 +8 x ¼ = 8 (x ¼ ) 4 = (8) 4 x = 4096

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c. x ½ – 6x ¼ – 16 = 0 (x ¼ ) 2 – 6(x ¼ ) – 16 = 0 ()() = 0 (x ¼ – 8)(x ¼ + 2) = 0 x ¼ – 8 = 0 OR x ¼ + 2 = 0 ( )( ) = 0 ()( ) = 0 x ¼ – 8 = 0 OR x ¼ + 2 = 0 +8 +8 -2 -2 x ¼ = 8 (x ¼ ) 4 = (8) 4 x = 4096

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c. x ½ – 6x ¼ – 16 = 0 (x ¼ ) 2 – 6(x ¼ ) – 16 = 0 ()() = 0 (x ¼ – 8)(x ¼ + 2) = 0 x ¼ – 8 = 0 OR x ¼ + 2 = 0 ( )( ) = 0 ()( ) = 0 x ¼ – 8 = 0 OR x ¼ + 2 = 0 +8 +8 -2 -2 x ¼ = 8 OR x ¼ = -2 (x ¼ ) 4 = (8) 4 x = 4096

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c. x ½ – 6x ¼ – 16 = 0 (x ¼ ) 2 – 6(x ¼ ) – 16 = 0 ()() = 0 (x ¼ – 8)(x ¼ + 2) = 0 x ¼ – 8 = 0 OR x ¼ + 2 = 0 ( )( ) = 0 ()( ) = 0 x ¼ – 8 = 0 OR x ¼ + 2 = 0 +8 +8 -2 -2 x ¼ = 8 OR x ¼ = -2 (x ¼ ) 4 = (8) 4 OR (x ¼ ) 4 = (-2) 4 x = 4096

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c. x ½ – 6x ¼ – 16 = 0 (x ¼ ) 2 – 6(x ¼ ) – 16 = 0 ()() = 0 (x ¼ – 8)(x ¼ + 2) = 0 x ¼ – 8 = 0 OR x ¼ + 2 = 0 ( )( ) = 0 ()( ) = 0 x ¼ – 8 = 0 OR x ¼ + 2 = 0 +8 +8 -2 -2 x ¼ = 8 OR x ¼ = -2 (x ¼ ) 4 = (8) 4 OR (x ¼ ) 4 = (-2) 4 x = 4096 OR x = 16

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c. x ½ – 6x ¼ – 16 = 0 (x ¼ ) 2 – 6(x ¼ ) – 16 = 0 ()() = 0 (x ¼ – 8)(x ¼ + 2) = 0 x ¼ – 8 = 0 OR x ¼ + 2 = 0 ( )( ) = 0 ()( ) = 0 x ¼ – 8 = 0 OR x ¼ + 2 = 0 +8 +8 -2 -2 x ¼ = 8 OR x ¼ = -2 (x ¼ ) 4 = (8) 4 OR (x ¼ ) 4 = (-2) 4 x = 4096 OR x = 16

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