Chapters

Solve the Biquadratic Equations
Exercise 1
Exercise 2
Exercise 3
Exercise 4
Exercise 5
Exercise 6
Solution of exercise 1
Solution of exercise 2
Solution of exercise 3
Solution of exercise 4
Solution of exercise 5
Solution of exercise 6

Solve the Biquadratic Equations

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Exercise 1

${ x }^{ 4 }-10{ x }^{ 2 }+9=0$

Exercise 2

${ x }^{ 4 }-13{ x }^{ 2 }+36=0$

Exercise 3

${ x }^{ 4 }-61{ x }^{ 2 }+900=0$

Exercise 4

${ x }^{ 4 }-25{ x }^{ 2 }+144=0$

Exercise 5

${ x }^{ 4 }-16{ x }^{ 2 }-225=0$

Exercise 6

${ x }^{ 6 }-7{ x }^{ 3 }+6=0$

Solution of exercise 1

${ x }^{ 4 }-10{ x }^{ 2 }+9=0$

Let ${ x }^{ 2 }=t$ :

${ t }^{ 2 }-10t+9=0$

$t=\frac { -b\pm \sqrt { { b }^{ 2 }-4ac } }{ 2a }$

$t=\frac { -(-10)\pm \sqrt { { (-10) }^{ 2 }-4(1)(9) } }{ 2(1) }$

$t=\frac { 10\pm \sqrt { 100-36 } }{ 2 }$

$t=\frac { 10\pm \sqrt { 64 } }{ 2 }$

$t=\frac { 10\pm 8 }{ 2 }$

$t=\frac { 10 + 8 }{ 2 } \qquad t=\frac { 10 - 8 }{ 2 }$

$t=\frac { 18 }{ 2 } \qquad t=\frac { 2 }{ 2 }$

$t=9 \qquad t=1$

${ x }^{ 2 }=9 \qquad { x }^{ 2 }=1$

$x=\pm3 \qquad x=\pm1$

Solution of exercise 2

${ x }^{ 4 }-13{ x }^{ 2 }+36=0$

Let ${ x }^{ 2 }=t$ :

${ t }^{ 2 }-13t+36=0$

$t=\frac { -b\pm \sqrt { { b }^{ 2 }-4ac } }{ 2a }$

$t=\frac { -(-13)\pm \sqrt { { (-13) }^{ 2 }-4(1)(36) } }{ 2(1) }$

$t=\frac { 13\pm \sqrt { 169-144 } }{ 2 }$

$t=\frac { 13\pm \sqrt { 25 } }{ 2 }$

$t=\frac { 13\pm 5 }{ 2 }$

$t=\frac { 13 + 5 }{ 2 } \qquad t=\frac { 13 - 5 }{ 2 }$

$t=\frac { 18 }{ 2 } \qquad t=\frac { 8 }{ 2 }$

$t=9 \qquad t=4$

${ x }^{ 2 }=9 \qquad { x }^{ 2 }=4$

$x=\pm 3 \qquad x=\pm 2$

Solution of exercise 3

${ x }^{ 4 }-61{ x }^{ 2 }+900=0$

Let ${ x }^{ 2 }=t$ :

${ t }^{ 2 }-61t+900=0$

$t=\frac { -b\pm \sqrt { { b }^{ 2 }-4ac } }{ 2a }$

$t=\frac { -(-61)\pm \sqrt { { (-61) }^{ 2 }-4(1)(900) } }{ 2(1) }$

$t=\frac { 61\pm \sqrt { 3721-3600 } }{ 2 }$

$t=\frac { 61\pm \sqrt { 121 } }{ 2 }$

$t=\frac { 61\pm 11 }{ 2 }$

$t=\frac { 61 + 11 }{ 2 } \qquad t=\frac { 61 - 11 }{ 2 }$

$t=\frac { 72 }{ 2 } \qquad t=\frac { 50 }{ 2 }$

$t=36 \qquad t=25$

${ x }^{ 2 }=36 \qquad { x }^{ 2 }=25$

$x= \pm 6 \qquad x= \pm 5$

Solution of exercise 4

${ x }^{ 4 }-25{ x }^{ 2 }+144=0$

Let ${ x }^{ 2 }=t$ :

${ t }^{ 2 }-25t+144=0$

$t=\frac { -b\pm \sqrt { { b }^{ 2 }-4ac } }{ 2a }$

$t=\frac { -(-25)\pm \sqrt { { (-25) }^{ 2 }-4(1)(144) } }{ 2(1) }$

$t=\frac { 25\pm \sqrt { 625-576 } }{ 2 }$

$t=\frac { 25\pm \sqrt { 49 } }{ 2 }$

$t=\frac { 25\pm 7 }{ 2 }$

$t=\frac { 25 + 7 }{ 2 } \qquad t=\frac { 25 - 7 }{ 2 }$

$t=\frac { 32 }{ 2 } \qquad t=\frac { 18 }{ 2 }$

$t=16 \qquad t=9$

${ x }^{ 2 } = 16 \qquad { x }^{ 2 }=9$

$x = \pm 4 \qquad x = \pm 3$

Solution of exercise 5

${ x }^{ 4 }-16{ x }^{ 2 }-225=0$

Let ${ x }^{ 2 }=t$ :

${ t }^{ 2 }-16t-225=0$

$t=\frac { -b\pm \sqrt { { b }^{ 2 }-4ac } }{ 2a }$

$t=\frac { -(-16)\pm \sqrt { { (-16) }^{ 2 }-4(1)(-225) } }{ 2(1) }$

$t=\frac { 16\pm \sqrt { 256+900 } }{ 2 }$

$t=\frac { 16\pm \sqrt { 1156 } }{ 2 }$

$t=\frac { 16\pm 34 }{ 2 }$

$t=\frac { 16 + 34 }{ 2 } \qquad t=\frac { 16 - 34 }{ 2 }$

$t=\frac { 50 }{ 2 } \qquad t=\frac { -18 }{ 2 }$

$t=25 \qquad t=-9$

${ x }^{ 2 } = 25 \qquad { x }^{ 2 } = -9$

$x= \pm 5 \qquad x= \pm \sqrt { -9 }\notin R$

Solution of exercise 6

${ x }^{ 6 }-7{ x }^{ 3 }+6=0$

Let ${ x }^{ 3 }=t$ :

${ t }^{ 2 }-7t+6=0$

$t=\frac { -b\pm \sqrt { { b }^{ 2 }-4ac } }{ 2a }$

$t=\frac { -(-7)\pm \sqrt { { (-7) }^{ 2 }-4(1)(6) } }{ 2(1) }$

$t=\frac { 7\pm \sqrt { 49-24 } }{ 2 }$

$t=\frac { 7\pm \sqrt { 25 } }{ 2 }$

$t=\frac { 7\pm 5 }{ 2 }$

$t=\frac { 7 + 5 }{ 2 } \qquad t=\frac { 7 - 5 }{ 2 }$

$t=\frac { 12 }{ 2 } \qquad t=\frac { 2 }{ 2 }$

$t=6 \qquad t=1$

${ x }^{ 3 }=6 \qquad { x }^{ 3 }=1$

$x=\sqrt [ 3 ]{ 6 } \qquad x=1$

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Biquadratic Equations Worksheet

Solve the Biquadratic Equations

Exercise 1

Exercise 2

Exercise 3

Exercise 4

Exercise 5

Exercise 6

Solution of exercise 1

Solution of exercise 2

Solution of exercise 3

Solution of exercise 4

Solution of exercise 5

Solution of exercise 6

Theory

Biquadratic Equations

Solving Systems of Equations

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Homogeneous Systems

Factoring Quadratic Equations

Linear Systems

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Linear System of Equations with Parameters Exercises

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Cramer’s Rule

Gauss Elimination Method

Solving Equations

Radical Equations

Solutions to Quadratic Equations

Quadratic Equations

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Cramer’s Rule

Quadratic Formula

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