Algebra Examples

$f (x) = 2 x^{4} - 9 x^{2} + 3$

Step 1

Set $2 x^{4} - 9 x^{2} + 3$ equal to $0$ .

$2 x^{4} - 9 x^{2} + 3 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Substitute $u = x^{2}$ into the equation. This will make the quadratic formula easy to use.

$2 u^{2} - 9 u + 3 = 0$

$u = x^{2}$

Step 2.2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2.3

Substitute the values $a = 2$ , $b = - 9$ , and $c = 3$ into the quadratic formula and solve for $u$ .

$\frac{9 \pm \sqrt{{(- 9)}^{2} - 4 \cdot (2 \cdot 3)}}{2 \cdot 2}$

Step 2.4

Simplify.

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Step 2.4.1

Simplify the numerator.

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Step 2.4.1.1

Raise $- 9$ to the power of $2$ .

$u = \frac{9 \pm \sqrt{81 - 4 \cdot 2 \cdot 3}}{2 \cdot 2}$

Step 2.4.1.2

Multiply $- 4 \cdot 2 \cdot 3$ .

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Step 2.4.1.2.1

Multiply $- 4$ by $2$ .

$u = \frac{9 \pm \sqrt{81 - 8 \cdot 3}}{2 \cdot 2}$

Step 2.4.1.2.2

Multiply $- 8$ by $3$ .

$u = \frac{9 \pm \sqrt{81 - 24}}{2 \cdot 2}$

Step 2.4.1.3

Subtract $24$ from $81$ .

$u = \frac{9 \pm \sqrt{57}}{2 \cdot 2}$

Step 2.4.2

Multiply $2$ by $2$ .

$u = \frac{9 \pm \sqrt{57}}{4}$

Step 2.5

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

Raise $- 9$ to the power of $2$ .

$u = \frac{9 \pm \sqrt{81 - 4 \cdot 2 \cdot 3}}{2 \cdot 2}$

Step 2.5.1.2

Multiply $- 4 \cdot 2 \cdot 3$ .

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Step 2.5.1.2.1

Multiply $- 4$ by $2$ .

$u = \frac{9 \pm \sqrt{81 - 8 \cdot 3}}{2 \cdot 2}$

Step 2.5.1.2.2

Multiply $- 8$ by $3$ .

$u = \frac{9 \pm \sqrt{81 - 24}}{2 \cdot 2}$

Step 2.5.1.3

Subtract $24$ from $81$ .

$u = \frac{9 \pm \sqrt{57}}{2 \cdot 2}$

Step 2.5.2

Multiply $2$ by $2$ .

$u = \frac{9 \pm \sqrt{57}}{4}$

Step 2.5.3

Change the $\pm$ to $+$ .

$u = \frac{9 + \sqrt{57}}{4}$

Step 2.6

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 2.6.1

Simplify the numerator.

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Step 2.6.1.1

Raise $- 9$ to the power of $2$ .

$u = \frac{9 \pm \sqrt{81 - 4 \cdot 2 \cdot 3}}{2 \cdot 2}$

Step 2.6.1.2

Multiply $- 4 \cdot 2 \cdot 3$ .

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Step 2.6.1.2.1

Multiply $- 4$ by $2$ .

$u = \frac{9 \pm \sqrt{81 - 8 \cdot 3}}{2 \cdot 2}$

Step 2.6.1.2.2

Multiply $- 8$ by $3$ .

$u = \frac{9 \pm \sqrt{81 - 24}}{2 \cdot 2}$

Step 2.6.1.3

Subtract $24$ from $81$ .

$u = \frac{9 \pm \sqrt{57}}{2 \cdot 2}$

Step 2.6.2

Multiply $2$ by $2$ .

$u = \frac{9 \pm \sqrt{57}}{4}$

Step 2.6.3

Change the $\pm$ to $-$ .

$u = \frac{9 - \sqrt{57}}{4}$

Step 2.7

The final answer is the combination of both solutions.

$u = \frac{9 + \sqrt{57}}{4}$ (Multiplicity of $1$ )

$u = \frac{9 - \sqrt{57}}{4}$ (Multiplicity of $1$ )

Step 2.8

Substitute the real value of $u = x^{2}$ back into the solved equation.

$x^{2} = 4.1374586$

${(x^{2})}^{1} = 0.36254139$

Step 2.9

Solve the first equation for $x$ .

$x^{2} = 4.1374586$

Step 2.10

Solve the equation for $x$ .

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Step 2.10.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{4.1374586}$

Step 2.10.2

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.10.2.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{4.1374586}$

Step 2.10.2.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{4.1374586}$

Step 2.10.2.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{4.1374586}, - \sqrt{4.1374586}$

Step 2.10.3

The multiplicity of a root is the number of times the root appears. For example, a factor of ${(x + 5)}^{3}$ would have a root at $x = - 5$ with multiplicity of $3$ .

$x = \sqrt{4.1374586}$ (Multiplicity of $1$ )

$x = - \sqrt{4.1374586}$ (Multiplicity of $1$ )

$x = \sqrt{4.1374586}$ (Multiplicity of $1$ )

$x = - \sqrt{4.1374586}$ (Multiplicity of $1$ )

Step 2.11

Solve the second equation for $x$ .

${(x^{2})}^{1} = 0.36254139$

Step 2.12

Solve the equation for $x$ .

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Step 2.12.1

Remove parentheses.

$x^{2} = 0.36254139$

Step 2.12.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0.36254139}$

Step 2.12.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.12.3.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{0.36254139}$

Step 2.12.3.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{0.36254139}$

Step 2.12.3.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{0.36254139}, - \sqrt{0.36254139}$

Step 2.12.4

The multiplicity of a root is the number of times the root appears. For example, a factor of ${(x + 5)}^{3}$ would have a root at $x = - 5$ with multiplicity of $3$ .

$x = \sqrt{0.36254139}$ (Multiplicity of $1$ )

$x = - \sqrt{0.36254139}$ (Multiplicity of $1$ )

$x = \sqrt{0.36254139}$ (Multiplicity of $1$ )

$x = - \sqrt{0.36254139}$ (Multiplicity of $1$ )

Step 2.13

The solution to $2 x^{4} - 9 x^{2} + 3 = 0$ is $x = \sqrt{4.1374586}, - \sqrt{4.1374586}, \sqrt{0.36254139}, - \sqrt{0.36254139}$ .

$x = \sqrt{4.1374586}, - \sqrt{4.1374586}, \sqrt{0.36254139}, - \sqrt{0.36254139}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \sqrt{4.1374586}, - \sqrt{4.1374586}, \sqrt{0.36254139}, - \sqrt{0.36254139}$

Decimal Form:

$x = 2.03407438 \dots, - 2.03407438 \dots, 0.60211410 \dots, - 0.60211410 \dots$

Step 4