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6.5 – Solving Equations with Quadratic Techniques
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Quadratic equations are in the form:
ax2 + bx + c,
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Quadratic equations are in the form:
ax2 + bx + c, where a, b, & c are integers
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Quadratic equations are in the form:
ax2 + bx + c, where a, b, & c are integers
exs.
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Quadratic equations are in the form:
ax2 + bx + c, where a, b, & c are integers
exs.
• x2 + 5x + 2
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Quadratic equations are in the form:
ax2 + bx + c, where a, b, & c are integers
exs.
• x2 + 5x + 2
• 2x2 – 18x + 13
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Quadratic equations are in the form:
ax2 + bx + c, where a, b, & c are integers
exs.
• x2 + 5x + 2
• 2x2 – 18x + 13
• x2 – 9
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Quadratic equations are in the form:
ax2 + bx + c, where a, b, & c are integers
exs.
• x2 + 5x + 2
• 2x2 – 18x + 13
• x2 – 9
x2 + 0x – 9
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Quadratic equations are in the form:ax2 + bx + c, where a, b, & c are integersexs.
• x2 + 5x + 2• 2x2 – 18x + 13• x2 – 9
x2 + 0x – 9• 2x2 + 8x
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Quadratic equations are in the form:ax2 + bx + c, where a, b, & c are integersexs.
• x2 + 5x + 2• 2x2 – 18x + 13• x2 – 9
x2 + 0x – 9• 2x2 + 8x
2x2 + 8x + 0
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Quadratic equations are in the form:ax2 + bx + c, where a, b, & c are integersexs.
• x2 + 5x + 2• 2x2 – 18x + 13• x2 – 9
x2 + 0x – 9• 2x2 + 8x
2x2 + 8x + 0NOTE: Must have the “x2” term to be a quadratic!
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Ex. 1 Write each expression in quadratic form, if possible.
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Ex. 1 Write each expression in quadratic form, if possible.
a. x4 + 13x2 + 36
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Ex. 1 Write each expression in quadratic form, if possible.
a. x4 + 13x2 + 36
(x2)2
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Ex. 1 Write each expression in quadratic form, if possible.
a. x4 + 13x2 + 36
(x2)2 + 13(x2)
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Ex. 1 Write each expression in quadratic form, if possible.
a. x4 + 13x2 + 36
(x2)2 + 13(x2) + 36
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Ex. 1 Write each expression in quadratic form, if possible.
a. x4 + 13x2 + 36
(x2)2 + 13(x2) + 36
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Ex. 1 Write each expression in quadratic form, if possible.
a. x4 + 13x2 + 36
(x2)2 + 13(x2) + 36
b. 16x6 – 625
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Ex. 1 Write each expression in quadratic form, if possible.
a. x4 + 13x2 + 36
(x2)2 + 13(x2) + 36
b. 16x6 – 625
(4x3)2
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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
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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. x½ – 9x¼ + 16
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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. x½ – 9x¼ + 16 (x¼)2
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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. x½ – 9x¼ + 16 (x¼)2 – 9(x¼)
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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. x½ – 9x¼ + 16 (x¼)2 – 9(x¼) + 16
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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. x½ – 9x¼ + 16 (x¼)2 – 9(x¼) + 16
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Ex. 2 Solve each equation.a. x4 = 16
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Ex. 2 Solve each equation.a. x4 = 16
-16 -16
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Ex. 2 Solve each equation.a. x4 = 16
-16 -16 x4 – 16 = 0
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Ex. 2 Solve each equation.a. x4 = 16
-16 -16 x4 – 16 = 0 (x2)2 – 16 = 0
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Ex. 2 Solve each equation.a. x4 = 16
-16 -16 x4 – 16 = 0 (x2)2 – 16 = 0
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Ex. 2 Solve each equation.a. x4 = 16
-16 -16 x4 – 16 = 0 (x2)2 – 16 = 0
( )( ) = 0
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Ex. 2 Solve each equation.a. x4 = 16
-16 -16 x4 – 16 = 0 (x2)2 – 16 = 0
( )( ) = 0(x2 )(x2 ) = 0
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Ex. 2 Solve each equation.a. x4 = 16
-16 -16 x4 – 16 = 0 (x2)2 – 16 = 0
( )( ) = 0(x2 – 4)(x2 + 4) = 0
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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 = 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. 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 = 0
x – 2 = 0 OR x + 2 = 0 OR x2 = -4 +2 +2 -2 -2
x = 2 OR x = -2
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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 = 0
x – 2 = 0 OR x + 2 = 0 OR x2 = -4 +2 +2 -2 -2 √x2 = √-4
x = 2 OR x = -2
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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 = 0
x – 2 = 0 OR x + 2 = 0 OR x2 = -4 +2 +2 -2 -2 √x2 = √-4
x = 2 OR x = -2 OR x = ±2i
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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 = 0
x – 2 = 0 OR x + 2 = 0 OR x2 = -4 +2 +2 -2 -2 √x2 = √-4
x = 2 OR x = -2 OR x = ±2i
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b. x4 + 11x2 + 18 = 0
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b. x4 + 11x2 + 18 = 0
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b. x4 + 11x2 + 18 = 0
(x2)2 + 11(x2) + 18 = 0
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b. x4 + 11x2 + 18 = 0
(x2)2 + 11(x2) + 18 = 0
( )( ) = 0
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b. x4 + 11x2 + 18 = 0
(x2)2 + 11(x2) + 18 = 0
( )( ) = 0
(x2 )(x2 ) = 0
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b. x4 + 11x2 + 18 = 0
(x2)2 + 11(x2) + 18 = 0
( )( ) = 0
(x2 + 9)(x2 + 2) = 0
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b. x4 + 11x2 + 18 = 0
(x2)2 + 11(x2) + 18 = 0
( )( ) = 0
(x2 + 9)(x2 + 2) = 0
x2 + 9 = 0
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b. x4 + 11x2 + 18 = 0
(x2)2 + 11(x2) + 18 = 0
( )( ) = 0
(x2 + 9)(x2 + 2) = 0
x2 + 9 = 0 OR x2 + 2 = 0
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b. x4 + 11x2 + 18 = 0
(x2)2 + 11(x2) + 18 = 0
( )( ) = 0
(x2 + 9)(x2 + 2) = 0
x2 + 9 = 0 OR x2 + 2 = 0
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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
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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
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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
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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
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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
![Page 63: 6.5 – Solving Equations with Quadratic Techniques](https://reader035.vdocument.in/reader035/viewer/2022062422/568135d8550346895d9d4771/html5/thumbnails/63.jpg)
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
![Page 64: 6.5 – Solving Equations with Quadratic Techniques](https://reader035.vdocument.in/reader035/viewer/2022062422/568135d8550346895d9d4771/html5/thumbnails/64.jpg)
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
x = ±3i
![Page 65: 6.5 – Solving Equations with Quadratic Techniques](https://reader035.vdocument.in/reader035/viewer/2022062422/568135d8550346895d9d4771/html5/thumbnails/65.jpg)
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
x = ±3i
![Page 66: 6.5 – Solving Equations with Quadratic Techniques](https://reader035.vdocument.in/reader035/viewer/2022062422/568135d8550346895d9d4771/html5/thumbnails/66.jpg)
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 (x + )(x + ) = 0 x2 + 9 = 0 -9 -9
x2 = -9 √x2 = √-9
x = ±3i
![Page 67: 6.5 – Solving Equations with Quadratic Techniques](https://reader035.vdocument.in/reader035/viewer/2022062422/568135d8550346895d9d4771/html5/thumbnails/67.jpg)
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 OR x2 + 2 = 0 -9 -9
x2 = -9 √x2 = √-9
x = ±3i
![Page 68: 6.5 – Solving Equations with Quadratic Techniques](https://reader035.vdocument.in/reader035/viewer/2022062422/568135d8550346895d9d4771/html5/thumbnails/68.jpg)
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 OR x2 + 2 = 0 -9 -9 -2 -2
x2 = -9 √x2 = √-9
x = ±3i
![Page 69: 6.5 – Solving Equations with Quadratic Techniques](https://reader035.vdocument.in/reader035/viewer/2022062422/568135d8550346895d9d4771/html5/thumbnails/69.jpg)
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 OR x2 + 2 = 0 -9 -9 -2 -2
x2 = -9 OR x2 = -2 √x2 = √-9
x = ±3i
![Page 70: 6.5 – Solving Equations with Quadratic Techniques](https://reader035.vdocument.in/reader035/viewer/2022062422/568135d8550346895d9d4771/html5/thumbnails/70.jpg)
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 OR x2 + 2 = 0 -9 -9 -2 -2
x2 = -9 OR x2 = -2 √x2 = √-9 OR √x2 = √-2
x = ±3i
![Page 71: 6.5 – Solving Equations with Quadratic Techniques](https://reader035.vdocument.in/reader035/viewer/2022062422/568135d8550346895d9d4771/html5/thumbnails/71.jpg)
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 OR x2 + 2 = 0 -9 -9 -2 -2
x2 = -9 OR x2 = -2 √x2 = √-9 OR √x2 = √-2
x = ±3i OR x = ±i√2
![Page 72: 6.5 – Solving Equations with Quadratic Techniques](https://reader035.vdocument.in/reader035/viewer/2022062422/568135d8550346895d9d4771/html5/thumbnails/72.jpg)
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 OR x2 + 2 = 0 -9 -9 -2 -2
x2 = -9 OR x2 = -2 √x2 = √-9 OR √x2 = √-2
x = ±3i OR x = ±i√2
![Page 73: 6.5 – Solving Equations with Quadratic Techniques](https://reader035.vdocument.in/reader035/viewer/2022062422/568135d8550346895d9d4771/html5/thumbnails/73.jpg)
c. x½ – 6x¼ – 16 = 0
![Page 74: 6.5 – Solving Equations with Quadratic Techniques](https://reader035.vdocument.in/reader035/viewer/2022062422/568135d8550346895d9d4771/html5/thumbnails/74.jpg)
c. x½ – 6x¼ – 16 = 0
![Page 75: 6.5 – Solving Equations with Quadratic Techniques](https://reader035.vdocument.in/reader035/viewer/2022062422/568135d8550346895d9d4771/html5/thumbnails/75.jpg)
c. x½ – 6x¼ – 16 = 0
(x¼)2 – 6(x¼) – 16 = 0
![Page 76: 6.5 – Solving Equations with Quadratic Techniques](https://reader035.vdocument.in/reader035/viewer/2022062422/568135d8550346895d9d4771/html5/thumbnails/76.jpg)
c. x½ – 6x¼ – 16 = 0
(x¼)2 – 6(x¼) – 16 = 0
( )( ) = 0
![Page 77: 6.5 – Solving Equations with Quadratic Techniques](https://reader035.vdocument.in/reader035/viewer/2022062422/568135d8550346895d9d4771/html5/thumbnails/77.jpg)
c. x½ – 6x¼ – 16 = 0
(x¼)2 – 6(x¼) – 16 = 0
( )( ) = 0
(x¼ )(x¼ ) = 0
![Page 78: 6.5 – Solving Equations with Quadratic Techniques](https://reader035.vdocument.in/reader035/viewer/2022062422/568135d8550346895d9d4771/html5/thumbnails/78.jpg)
c. x½ – 6x¼ – 16 = 0
(x¼)2 – 6(x¼) – 16 = 0
( )( ) = 0
(x¼ – 8)(x¼ + 2) = 0
![Page 79: 6.5 – Solving Equations with Quadratic Techniques](https://reader035.vdocument.in/reader035/viewer/2022062422/568135d8550346895d9d4771/html5/thumbnails/79.jpg)
c. x½ – 6x¼ – 16 = 0
(x¼)2 – 6(x¼) – 16 = 0
( )( ) = 0
(x¼ – 8)(x¼ + 2) = 0
x¼ – 8 = 0
![Page 80: 6.5 – Solving Equations with Quadratic Techniques](https://reader035.vdocument.in/reader035/viewer/2022062422/568135d8550346895d9d4771/html5/thumbnails/80.jpg)
c. x½ – 6x¼ – 16 = 0
(x¼)2 – 6(x¼) – 16 = 0
( )( ) = 0
(x¼ – 8)(x¼ + 2) = 0
x¼ – 8 = 0 OR x¼ + 2 = 0
![Page 81: 6.5 – Solving Equations with Quadratic Techniques](https://reader035.vdocument.in/reader035/viewer/2022062422/568135d8550346895d9d4771/html5/thumbnails/81.jpg)
c. x½ – 6x¼ – 16 = 0
(x¼)2 – 6(x¼) – 16 = 0
( )( ) = 0
(x¼ – 8)(x¼ + 2) = 0
x¼ – 8 = 0 OR x¼ + 2 = 0
( )( ) = 0
![Page 82: 6.5 – Solving Equations with Quadratic Techniques](https://reader035.vdocument.in/reader035/viewer/2022062422/568135d8550346895d9d4771/html5/thumbnails/82.jpg)
c. x½ – 6x¼ – 16 = 0
(x¼)2 – 6(x¼) – 16 = 0
( )( ) = 0
(x¼ – 8)(x¼ + 2) = 0
x¼ – 8 = 0 OR x¼ + 2 = 0
( )( ) = 0
x¼ – 8 = 0
![Page 83: 6.5 – Solving Equations with Quadratic Techniques](https://reader035.vdocument.in/reader035/viewer/2022062422/568135d8550346895d9d4771/html5/thumbnails/83.jpg)
c. x½ – 6x¼ – 16 = 0
(x¼)2 – 6(x¼) – 16 = 0
( )( ) = 0
(x¼ – 8)(x¼ + 2) = 0
x¼ – 8 = 0 OR x¼ + 2 = 0
( )( ) = 0
x¼ – 8 = 0
+8 +8
![Page 84: 6.5 – Solving Equations with Quadratic Techniques](https://reader035.vdocument.in/reader035/viewer/2022062422/568135d8550346895d9d4771/html5/thumbnails/84.jpg)
c. x½ – 6x¼ – 16 = 0 (x¼)2 – 6(x¼) – 16 = 0( )( ) = 0(x¼ – 8)(x¼ + 2) = 0
x¼ – 8 = 0 OR x¼ + 2 = 0 ( )( ) = 0
x¼ – 8 = 0 +8 +8 x¼ = 8
![Page 85: 6.5 – Solving Equations with Quadratic Techniques](https://reader035.vdocument.in/reader035/viewer/2022062422/568135d8550346895d9d4771/html5/thumbnails/85.jpg)
c. x½ – 6x¼ – 16 = 0 (x¼)2 – 6(x¼) – 16 = 0( )( ) = 0(x¼ – 8)(x¼ + 2) = 0
x¼ – 8 = 0 OR x¼ + 2 = 0 ( )( ) = 0
x¼ – 8 = 0 +8 +8 x¼ = 8 (x¼)4 = (8)4
![Page 86: 6.5 – Solving Equations with Quadratic Techniques](https://reader035.vdocument.in/reader035/viewer/2022062422/568135d8550346895d9d4771/html5/thumbnails/86.jpg)
c. x½ – 6x¼ – 16 = 0 (x¼)2 – 6(x¼) – 16 = 0( )( ) = 0(x¼ – 8)(x¼ + 2) = 0
x¼ – 8 = 0 OR x¼ + 2 = 0 ( )( ) = 0
x¼ – 8 = 0 +8 +8 x¼ = 8 (x¼)4 = (8)4 x = 4096
![Page 87: 6.5 – Solving Equations with Quadratic Techniques](https://reader035.vdocument.in/reader035/viewer/2022062422/568135d8550346895d9d4771/html5/thumbnails/87.jpg)
c. x½ – 6x¼ – 16 = 0 (x¼)2 – 6(x¼) – 16 = 0( )( ) = 0(x¼ – 8)(x¼ + 2) = 0
x¼ – 8 = 0 OR x¼ + 2 = 0 ( )( ) = 0
x¼ – 8 = 0 +8 +8 x¼ = 8 (x¼)4 = (8)4 x = 4096
![Page 88: 6.5 – Solving Equations with Quadratic Techniques](https://reader035.vdocument.in/reader035/viewer/2022062422/568135d8550346895d9d4771/html5/thumbnails/88.jpg)
c. x½ – 6x¼ – 16 = 0 (x¼)2 – 6(x¼) – 16 = 0( )( ) = 0(x¼ – 8)(x¼ + 2) = 0
x¼ – 8 = 0 OR x¼ + 2 = 0 ( )( ) = 0 ( )( ) = 0
x¼ – 8 = 0 +8 +8 x¼ = 8 (x¼)4 = (8)4 x = 4096
![Page 89: 6.5 – Solving Equations with Quadratic Techniques](https://reader035.vdocument.in/reader035/viewer/2022062422/568135d8550346895d9d4771/html5/thumbnails/89.jpg)
c. x½ – 6x¼ – 16 = 0 (x¼)2 – 6(x¼) – 16 = 0( )( ) = 0(x¼ – 8)(x¼ + 2) = 0
x¼ – 8 = 0 OR x¼ + 2 = 0 ( )( ) = 0 ( )( ) = 0
x¼ – 8 = 0 OR x¼ + 2 = 0 +8 +8 x¼ = 8 (x¼)4 = (8)4 x = 4096
![Page 90: 6.5 – Solving Equations with Quadratic Techniques](https://reader035.vdocument.in/reader035/viewer/2022062422/568135d8550346895d9d4771/html5/thumbnails/90.jpg)
c. x½ – 6x¼ – 16 = 0 (x¼)2 – 6(x¼) – 16 = 0( )( ) = 0(x¼ – 8)(x¼ + 2) = 0
x¼ – 8 = 0 OR x¼ + 2 = 0 ( )( ) = 0 ( )( ) = 0
x¼ – 8 = 0 OR x¼ + 2 = 0 +8 +8 -2 -2 x¼ = 8 (x¼)4 = (8)4 x = 4096
![Page 91: 6.5 – Solving Equations with Quadratic Techniques](https://reader035.vdocument.in/reader035/viewer/2022062422/568135d8550346895d9d4771/html5/thumbnails/91.jpg)
c. x½ – 6x¼ – 16 = 0 (x¼)2 – 6(x¼) – 16 = 0( )( ) = 0(x¼ – 8)(x¼ + 2) = 0
x¼ – 8 = 0 OR x¼ + 2 = 0 ( )( ) = 0 ( )( ) = 0
x¼ – 8 = 0 OR x¼ + 2 = 0 +8 +8 -2 -2 x¼ = 8 OR x¼ = -2 (x¼)4 = (8)4 x = 4096
![Page 92: 6.5 – Solving Equations with Quadratic Techniques](https://reader035.vdocument.in/reader035/viewer/2022062422/568135d8550346895d9d4771/html5/thumbnails/92.jpg)
c. x½ – 6x¼ – 16 = 0 (x¼)2 – 6(x¼) – 16 = 0( )( ) = 0(x¼ – 8)(x¼ + 2) = 0
x¼ – 8 = 0 OR x¼ + 2 = 0 ( )( ) = 0 ( )( ) = 0
x¼ – 8 = 0 OR x¼ + 2 = 0 +8 +8 -2 -2 x¼ = 8 OR x¼ = -2 (x¼)4 = (8)4 OR (x¼)4 = (-2)4 x = 4096
![Page 93: 6.5 – Solving Equations with Quadratic Techniques](https://reader035.vdocument.in/reader035/viewer/2022062422/568135d8550346895d9d4771/html5/thumbnails/93.jpg)
c. x½ – 6x¼ – 16 = 0 (x¼)2 – 6(x¼) – 16 = 0( )( ) = 0(x¼ – 8)(x¼ + 2) = 0
x¼ – 8 = 0 OR x¼ + 2 = 0 ( )( ) = 0 ( )( ) = 0
x¼ – 8 = 0 OR x¼ + 2 = 0 +8 +8 -2 -2 x¼ = 8 OR x¼ = -2 (x¼)4 = (8)4 OR (x¼)4 = (-2)4 x = 4096 OR x = 16
![Page 94: 6.5 – Solving Equations with Quadratic Techniques](https://reader035.vdocument.in/reader035/viewer/2022062422/568135d8550346895d9d4771/html5/thumbnails/94.jpg)
c. x½ – 6x¼ – 16 = 0 (x¼)2 – 6(x¼) – 16 = 0( )( ) = 0(x¼ – 8)(x¼ + 2) = 0
x¼ – 8 = 0 OR x¼ + 2 = 0 ( )( ) = 0 ( )( ) = 0
x¼ – 8 = 0 OR x¼ + 2 = 0 +8 +8 -2 -2 x¼ = 8 OR x¼ = -2 (x¼)4 = (8)4 OR (x¼)4 = (-2)4 x = 4096 OR x = 16