Finite Math Examples

$y = 2 x - x^{3}$

Step 1

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

$2 x - x^{3} = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor $x$ out of $2 x - x^{3}$ .

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Step 2.1.1

Factor $x$ out of $2 x$ .

$x \cdot 2 - x^{3} = 0$

Step 2.1.2

Factor $x$ out of $- x^{3}$ .

$x \cdot 2 + x (- x^{2}) = 0$

Step 2.1.3

Factor $x$ out of $x \cdot 2 + x (- x^{2})$ .

$x (2 - x^{2}) = 0$

Step 2.2

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

$x = 0$

$2 - x^{2} = 0$

Step 2.3

Set $x$ equal to $0$ .

$x = 0$

Step 2.4

Set $2 - x^{2}$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $2 - x^{2}$ equal to $0$ .

$2 - x^{2} = 0$

Step 2.4.2

Solve $2 - x^{2} = 0$ for $x$ .

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Step 2.4.2.1

Subtract $2$ from both sides of the equation.

$- x^{2} = - 2$

Step 2.4.2.2

Divide each term in $- x^{2} = - 2$ by $- 1$ and simplify.

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Step 2.4.2.2.1

Divide each term in $- x^{2} = - 2$ by $- 1$ .

$\frac{- x^{2}}{- 1} = \frac{- 2}{- 1}$

Step 2.4.2.2.2

Simplify the left side.

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Step 2.4.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} = \frac{- 2}{- 1}$

Step 2.4.2.2.2.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{- 2}{- 1}$

Step 2.4.2.2.3

Simplify the right side.

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Step 2.4.2.2.3.1

Divide $- 2$ by $- 1$ .

$x^{2} = 2$

Step 2.4.2.3

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

$x = \pm \sqrt{2}$

Step 2.4.2.4

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

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Step 2.4.2.4.1

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

$x = \sqrt{2}$

Step 2.4.2.4.2

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

$x = - \sqrt{2}$

Step 2.4.2.4.3

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

$x = \sqrt{2}, - \sqrt{2}$

Step 2.5

The final solution is all the values that make $x (2 - x^{2}) = 0$ true.

$x = 0, \sqrt{2}, - \sqrt{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = 0, \sqrt{2}, - \sqrt{2}$

Decimal Form:

$x = 0, 1.41421356 \dots, - 1.41421356 \dots$

Step 4