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6.5 – Solving Equations with Quadratic Techniques. Quadratic equations are in the form: a x 2 + b x + c ,. Quadratic equations are in the form: a x 2 + b x + c , where a, b, & c are integers. Quadratic equations are in the form: a x 2 + b x + c , where a, b, & c are integers exs. . - PowerPoint PPT Presentation

6.5 Solving Equations with Quadratic Techniques

Quadratic equations are in the form:ax2 + bx + c,

Quadratic equations are in the form:ax2 + bx + c, where a, b, & c are integers

Quadratic equations are in the form:ax2 + bx + c, where a, b, & c are integersexs.

Quadratic equations are in the form:ax2 + bx + c, where a, b, & c are integersexs. x2 + 5x + 2

Quadratic equations are in the form:ax2 + bx + c, where a, b, & c are integersexs. x2 + 5x + 22x2 18x + 13

Quadratic equations are in the form:ax2 + bx + c, where a, b, & c are integersexs. x2 + 5x + 22x2 18x + 13x2 9

Quadratic equations are in the form:ax2 + bx + c, where a, b, & c are integersexs. x2 + 5x + 22x2 18x + 13x2 9x2 + 0x 9

Quadratic equations are in the form:ax2 + bx + c, where a, b, & c are integersexs. x2 + 5x + 22x2 18x + 13x2 9x2 + 0x 92x2 + 8x

Quadratic equations are in the form:ax2 + bx + c, where a, b, & c are integersexs. x2 + 5x + 22x2 18x + 13x2 9x2 + 0x 92x2 + 8x 2x2 + 8x + 0

Quadratic equations are in the form:ax2 + bx + c, where a, b, & c are integersexs. x2 + 5x + 22x2 18x + 13x2 9x2 + 0x 92x2 + 8x 2x2 + 8x + 0NOTE: Must have the x2 term to be a quadratic!

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

Ex. 1 Write each expression in quadratic form, if possible.a. x4 + 13x2 + 36

Ex. 1 Write each expression in quadratic form, if possible.a. x4 + 13x2 + 36 (x2)2

Ex. 1 Write each expression in quadratic form, if possible.a. x4 + 13x2 + 36 (x2)2 + 13(x2)

Ex. 1 Write each expression in quadratic form, if possible.a. x4 + 13x2 + 36 (x2)2 + 13(x2) + 36

Ex. 1 Write each expression in quadratic form, if possible.a. x4 + 13x2 + 36 (x2)2 + 13(x2) + 36

Ex. 1 Write each expression in quadratic form, if possible.a. x4 + 13x2 + 36 (x2)2 + 13(x2) + 36

b. 16x6 625

Ex. 1 Write each expression in quadratic form, if possible.a. x4 + 13x2 + 36 (x2)2 + 13(x2) + 36

b. 16x6 625 (4x3)2

Ex. 1 Write each expression in quadratic form, if possible.a. x4 + 13x2 + 36 (x2)2 + 13(x2) + 36

b. 16x6 625 (4x3)2 625

Ex. 1 Write each expression in quadratic form, if possible.a. x4 + 13x2 + 36 (x2)2 + 13(x2) + 36

b. 16x6 625 (4x3)2 625

c. x 9x + 16

Ex. 1 Write each expression in quadratic form, if possible.a. x4 + 13x2 + 36 (x2)2 + 13(x2) + 36

b. 16x6 625 (4x3)2 625

c. x 9x + 16 (x)2

Ex. 1 Write each expression in quadratic form, if possible.a. x4 + 13x2 + 36 (x2)2 + 13(x2) + 36

b. 16x6 625 (4x3)2 625

c. x 9x + 16 (x)2 9(x)

Ex. 1 Write each expression in quadratic form, if possible.a. x4 + 13x2 + 36 (x2)2 + 13(x2) + 36

b. 16x6 625 (4x3)2 625

c. x 9x + 16 (x)2 9(x) + 16

Ex. 1 Write each expression in quadratic form, if possible.a. x4 + 13x2 + 36 (x2)2 + 13(x2) + 36

b. 16x6 625 (4x3)2 625

c. x 9x + 16 (x)2 9(x) + 16

Ex. 2 Solve each equation.a. x4 = 16

Ex. 2 Solve each equation.a. x4 = 16 -16 -16

Ex. 2 Solve each equation.a. x4 = 16 -16 -16 x4 16 = 0

Ex. 2 Solve each equation.a. x4 = 16 -16 -16 x4 16 = 0 (x2)2 16 = 0

Ex. 2 Solve each equation.a. x4 = 16 -16 -16 x4 16 = 0 (x2)2 16 = 0

Ex. 2 Solve each equation.a. x4 = 16 -16 -16 x4 16 = 0 (x2)2 16 = 0 ()() = 0

Ex. 2 Solve each equation.a. x4 = 16 -16 -16 x4 16 = 0 (x2)2 16 = 0()() = 0(x2 )(x2 ) = 0

Ex. 2 Solve each equation.a. x4 = 16 -16 -16 x4 16 = 0 (x2)2 16 = 0()() = 0(x2 4)(x2 + 4) = 0

Ex. 2 Solve each equation.a. x4 = 16 -16 -16 x4 16 = 0 (x2)2 16 = 0()() = 0(x2 4)(x2 + 4) = 0 x2 4 = 0

Ex. 2 Solve each equation.a. x4 = 16 -16 -16 x4 16 = 0 (x2)2 16 = 0()() = 0(x2 4)(x2 + 4) = 0 x2 4 = 0 OR x2 + 4 = 0

Ex. 2 Solve each equation.a. x4 = 16 -16 -16 x4 16 = 0 (x2)2 16 = 0()() = 0(x2 4)(x2 + 4) = 0 x2 4 = 0 OR x2 + 4 = 0 ( )( ) = 0

Ex. 2 Solve each equation.a. x4 = 16 -16 -16 x4 16 = 0 (x2)2 16 = 0()() = 0(x2 4)(x2 + 4) = 0 x2 4 = 0 OR x2 + 4 = 0 ( )( ) = 0 (x )(x ) = 0

Ex. 2 Solve each equation.a. x4 = 16 -16 -16 x4 16 = 0 (x2)2 16 = 0()() = 0(x2 4)(x2 + 4) = 0 x2 4 = 0 OR x2 + 4 = 0 ( )( ) = 0 (x 2)(x + 2) = 0

Ex. 2 Solve each equation.a. x4 = 16 -16 -16 x4 16 = 0 (x2)2 16 = 0()() = 0(x2 4)(x2 + 4) = 0 x2 4 = 0 OR x2 + 4 = 0 ( )( ) = 0 (x 2)(x + 2) = 0 x 2 = 0 OR x + 2 = 0

Ex. 2 Solve each equation.a. x4 = 16 -16 -16 x4 16 = 0 (x2)2 16 = 0()() = 0(x2 4)(x2 + 4) = 0 x2 4 = 0 OR x2 + 4 = 0 ( )( ) = 0 (x 2)(x + 2) = 0 x 2 = 0 OR x + 2 = 0 +2 +2 -2 -2

Ex. 2 Solve each equation.a. x4 = 16 -16 -16 x4 16 = 0 (x2)2 16 = 0()() = 0(x2 4)(x2 + 4) = 0 x2 4 = 0 OR x2 + 4 = 0 ( )( ) = 0 (x 2)(x + 2) = 0 x 2 = 0 OR x + 2 = 0 +2 +2 -2 -2 x = 2 OR x = -2

Ex. 2 Solve each equation.a. x4 = 16 -16 -16 x4 16 = 0 (x2)2 16 = 0()() = 0(x2 4)(x2 + 4) = 0 x2 4 = 0 OR x2 + 4 = 0 ( )( ) = 0 (x 2)(x + 2) = 0 x 2 = 0 OR x + 2 = 0 +2 +2 -2 -2 x = 2 OR x = -2

Ex. 2 Solve each equation.a. x4 = 16 -16 -16 x4 16 = 0 (x2)2 16 = 0()() = 0(x2 4)(x2 + 4) = 0 x2 4 = 0 OR x2 + 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

Ex. 2 Solve each equation.a. x4 = 16 -16 -16 x4 16 = 0 (x2)2 16 = 0()() = 0(x2 4)(x2 + 4) = 0 x2 4 = 0 OR x2 + 4 = 0 ( )( ) = 0 OR ( )( ) = 0 (x 2)(x + 2) = 0 OR x2 + 4 = 0x 2 = 0 OR x + 2 = 0 +2 +2 -2 -2 x = 2 OR x = -2

Ex. 2 Solve each equation.a. x4 = 16 -16 -16 x4 16 = 0 (x2)2 16 = 0()() = 0(x2 4)(x2 + 4) = 0 x2 4 = 0 OR x2 + 4 = 0 ( )( ) = 0 OR ( )( ) = 0 (x 2)(x + 2) = 0 OR x2 + 4 = 0x 2 = 0 OR x + 2 = 0 OR x2 = -4 +2 +2 -2 -2 x = 2 OR x = -2

Ex. 2 Solve each equation.a. x4 = 16 -16 -16 x4 16 = 0 (x2)2 16 = 0()() = 0(x2 4)(x2 + 4) = 0 x2 4 = 0 OR x2 + 4 = 0 ( )( ) = 0 OR ( )( ) = 0 (x 2)(x + 2) = 0 OR x2 + 4 = 0x 2 = 0 OR x + 2 = 0 OR x2 = -4 +2 +2 -2 -2 x2 = -4 x = 2 OR x = -2

Ex. 2 Solve each equation.a. x4 = 16 -16 -16 x4 16 = 0 (x2)2 16 = 0()() = 0(x2 4)(x2 + 4) = 0 x2 4 = 0 OR x2 + 4 = 0 ( )( ) = 0 OR ( )( ) = 0 (x 2)(x + 2) = 0 OR x2 + 4 = 0x 2 = 0 OR x + 2 = 0 OR x2 = -4 +2 +2 -2 -2 x2 = -4 x = 2 OR x = -2 OR x = 2i

Ex. 2 Solve each equation.a. x4 = 16 -16 -16 x4 16 = 0 (x2)2 16 = 0()() = 0(x2 4)(x2 + 4) = 0 x2 4 = 0 OR x2 + 4 = 0 ( )( ) = 0 OR ( )( ) = 0 (x 2)(x + 2) = 0 OR x2 + 4 = 0x 2 = 0 OR x + 2 = 0 OR x2 = -4 +2 +2 -2 -2 x2 = -4 x = 2 OR x = -2 OR x = 2i

b. x4 + 11x2 + 18 = 0

b. x4 + 11x2 + 18 = 0

b. x4 + 11x2 + 18 = 0 (x2)2 + 11(x2) + 18 = 0

b. x4 + 11x2 + 18 = 0 (x2)2 + 11(x2) + 18 = 0()() = 0

b. x4 + 11x2 + 18 = 0 (x2)2 + 11(x2) + 18 = 0()() = 0 (x2 )(x2 ) = 0

b. x4 + 11x2 + 18 = 0 (x2)2 + 11(x2) + 18 = 0()() = 0 (x2 + 9)(x2 + 2) = 0

b. x4 + 11x2 + 18 = 0 (x2)2 + 11(x2) + 18 = 0()() = 0 (x2 + 9)(x2 + 2) = 0 x2 + 9 = 0

b. x4 + 11x2 + 18 = 0 (x2)2 + 11(x2) + 18 = 0()() = 0 (x2 + 9)(x2 + 2) = 0 x2 + 9 = 0 OR x2 + 2 = 0

b. x4 + 11x2 + 18 = 0 (x2)2 + 11(x2) + 18 = 0()() = 0 (x2 + 9)(x2 + 2) = 0 x2 + 9 = 0 OR x2 + 2 = 0

b. x4 + 11x2 + 18 = 0 (x2)2 + 11(x2) + 18 = 0()() = 0 (x2 + 9)(x2 + 2) = 0 x2 + 9 = 0 OR x2 + 2 = 0( )( ) = 0

b. x4 + 11x2 + 18 = 0 (x2)2 + 11(x2) + 18 = 0()() = 0 (x2 + 9)(x2 + 2) = 0 x2 + 9 = 0 OR x2 + 2 = 0(x + )(x + ) = 0

b. x4 + 11x2 + 18 = 0 (x2)2 + 11(x2) + 18 = 0()() = 0 (x2 + 9)(x2 + 2) = 0 x2 + 9 = 0 OR x2 + 2 = 0( )( ) = 0 x2 + 9 = 0

b. x4 + 11x2 + 18 = 0 (x2)2 + 11(x2) + 18 = 0()() = 0 (x2 + 9)(x2 + 2) = 0 x2 + 9 = 0 OR x2 + 2 = 0( )( ) = 0 ()() = 0 x2 + 9 = 0 -9 -9

b. x4 + 11x2 + 18 = 0 (x2)2 + 11(x2) + 18 = 0()() = 0 (x2 + 9)(x2 + 2) = 0 x2 + 9 = 0 OR x2 + 2 = 0( )( ) = 0 ()() = 0 x2 + 9 = 0 -9 -9 x2 = -9

b. x4 + 11x2 + 18 = 0 (x2)2 + 11(x2) + 18 = 0()() = 0 (x2 + 9)(x2 + 2) = 0 x2 + 9 = 0 OR x2 + 2 = 0( )( ) = 0 ()() = 0 x2 + 9 = 0 -9 -9 x2 = -9 x2 = -9

b. x4 + 11x2 + 18 = 0 (x2)2 + 11(x2) + 18 = 0()() = 0 (x2 + 9)(x2 + 2) = 0 x2 + 9 = 0 OR x2 + 2