Finite Math Examples

$f (x) = \frac{1}{4} x^{4} - 6 x^{3 + 7}$

Step 1

Set $\frac{1}{4} x^{4} - 6 x^{3 + 7}$ equal to $0$ .

$\frac{1}{4} x^{4} - 6 x^{3 + 7} = 0$

Step 2

Solve for $x$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Combine $\frac{1}{4}$ and $x^{4}$ .

$\frac{x^{4}}{4} - 6 x^{3 + 7} = 0$

Step 2.1.2

Add $3$ and $7$ .

$\frac{x^{4}}{4} - 6 x^{10} = 0$

Step 2.2

Multiply each term in $\frac{x^{4}}{4} - 6 x^{10} = 0$ by $4$ to eliminate the fractions.

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Step 2.2.1

Multiply each term in $\frac{x^{4}}{4} - 6 x^{10} = 0$ by $4$ .

$\frac{x^{4}}{4} \cdot 4 - 6 x^{10} \cdot 4 = 0 \cdot 4$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

Cancel the common factor of $4$ .

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Step 2.2.2.1.1.1

Cancel the common factor.

$\frac{x^{4}}{4} \cdot 4 - 6 x^{10} \cdot 4 = 0 \cdot 4$

Step 2.2.2.1.1.2

Rewrite the expression.

$x^{4} - 6 x^{10} \cdot 4 = 0 \cdot 4$

Step 2.2.2.1.2

Multiply $4$ by $- 6$ .

$x^{4} - 24 x^{10} = 0 \cdot 4$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Multiply $0$ by $4$ .

$x^{4} - 24 x^{10} = 0$

Step 2.3

Factor $x^{4}$ out of $x^{4} - 24 x^{10}$ .

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Step 2.3.1

Multiply by $1$ .

$x^{4} \cdot 1 - 24 x^{10} = 0$

Step 2.3.2

Factor $x^{4}$ out of $- 24 x^{10}$ .

$x^{4} \cdot 1 + x^{4} (- 24 x^{6}) = 0$

Step 2.3.3

Factor $x^{4}$ out of $x^{4} \cdot 1 + x^{4} (- 24 x^{6})$ .

$x^{4} (1 - 24 x^{6}) = 0$

Step 2.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x^{4} = 0$

$1 - 24 x^{6} = 0$

Step 2.5

Set $x^{4}$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $x^{4}$ equal to $0$ .

$x^{4} = 0$

Step 2.5.2

Solve $x^{4} = 0$ for $x$ .

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Step 2.5.2.1

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

$x = \pm \sqrt[4]{0}$

Step 2.5.2.2

Simplify $\pm \sqrt[4]{0}$ .

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Step 2.5.2.2.1

Rewrite $0$ as $0^{4}$ .

$x = \pm \sqrt[4]{0^{4}}$

Step 2.5.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 2.5.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 2.6

Set $1 - 24 x^{6}$ equal to $0$ and solve for $x$ .

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Step 2.6.1

Set $1 - 24 x^{6}$ equal to $0$ .

$1 - 24 x^{6} = 0$

Step 2.6.2

Solve $1 - 24 x^{6} = 0$ for $x$ .

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Step 2.6.2.1

Subtract $1$ from both sides of the equation.

$- 24 x^{6} = - 1$

Step 2.6.2.2

Divide each term in $- 24 x^{6} = - 1$ by $- 24$ and simplify.

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Step 2.6.2.2.1

Divide each term in $- 24 x^{6} = - 1$ by $- 24$ .

$\frac{- 24 x^{6}}{- 24} = \frac{- 1}{- 24}$

Step 2.6.2.2.2

Simplify the left side.

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Step 2.6.2.2.2.1

Cancel the common factor of $- 24$ .

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Step 2.6.2.2.2.1.1

Cancel the common factor.

$\frac{- 24 x^{6}}{- 24} = \frac{- 1}{- 24}$

Step 2.6.2.2.2.1.2

Divide $x^{6}$ by $1$ .

$x^{6} = \frac{- 1}{- 24}$

Step 2.6.2.2.3

Simplify the right side.

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Step 2.6.2.2.3.1

Dividing two negative values results in a positive value.

$x^{6} = \frac{1}{24}$

Step 2.6.2.3

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

$x = \pm \sqrt[6]{\frac{1}{24}}$

Step 2.6.2.4

Simplify $\pm \sqrt[6]{\frac{1}{24}}$ .

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Step 2.6.2.4.1

Rewrite $\sqrt[6]{\frac{1}{24}}$ as $\frac{\sqrt[6]{1}}{\sqrt[6]{24}}$ .

$x = \pm \frac{\sqrt[6]{1}}{\sqrt[6]{24}}$

Step 2.6.2.4.2

Any root of $1$ is $1$ .

$x = \pm \frac{1}{\sqrt[6]{24}}$

Step 2.6.2.5

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

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Step 2.6.2.5.1

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

$x = \frac{1}{\sqrt[6]{24}}$

Step 2.6.2.5.2

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

$x = - \frac{1}{\sqrt[6]{24}}$

Step 2.6.2.5.3

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

$x = \frac{1}{\sqrt[6]{24}}, - \frac{1}{\sqrt[6]{24}}$

Step 2.7

The final solution is all the values that make $x^{4} (1 - 24 x^{6}) = 0$ true.

$x = 0, \frac{1}{\sqrt[6]{24}}, - \frac{1}{\sqrt[6]{24}}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = 0, \frac{1}{\sqrt[6]{24}}, - \frac{1}{\sqrt[6]{24}}$

Decimal Form:

$x = 0, 0.58879592 \dots, - 0.58879592 \dots$

Step 4