6.5 – Solving Equations with Quadratic Techniques.

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Presentation transcript:

6.5 – Solving Equations with Quadratic Techniques

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

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

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

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

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

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

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

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

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

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!

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

a. x x

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

b. x x = 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

c. x ½ – 6x ¼ – 16 = 0

(x ¼ ) 2 – 6(x ¼ ) – 16 = 0

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

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

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

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

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

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

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

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 =

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 = x ¼ = 8

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 = x ¼ = 8 (x ¼ ) 4 = (8) 4

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 = x ¼ = 8 (x ¼ ) 4 = (8) 4 x = 4096

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 = x ¼ = 8 (x ¼ ) 4 = (8) 4 x = 4096

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 = x ¼ = 8 (x ¼ ) 4 = (8) 4 x = 4096

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 = x ¼ = 8 (x ¼ ) 4 = (8) 4 x = 4096

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 = x ¼ = 8 (x ¼ ) 4 = (8) 4 x = 4096

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 = x ¼ = 8 OR x ¼ = -2 (x ¼ ) 4 = (8) 4 x = 4096

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 = x ¼ = 8 OR x ¼ = -2 (x ¼ ) 4 = (8) 4 OR (x ¼ ) 4 = (-2) 4 x = 4096

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 = x ¼ = 8 OR x ¼ = -2 (x ¼ ) 4 = (8) 4 OR (x ¼ ) 4 = (-2) 4 x = 4096 OR x = 16

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 = x ¼ = 8 OR x ¼ = -2 (x ¼ ) 4 = (8) 4 OR (x ¼ ) 4 = (-2) 4 x = 4096 OR x = 16