Algebra Examples

$5 x^{3} = 5 x^{2} + 12 x$

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract $5 x^{2}$ from both sides of the equation.

$5 x^{3} - 5 x^{2} = 12 x$

Step 1.2

Subtract $12 x$ from both sides of the equation.

$5 x^{3} - 5 x^{2} - 12 x = 0$

Step 2

Factor $x$ out of $5 x^{3} - 5 x^{2} - 12 x$ .

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Step 2.1

Factor $x$ out of $5 x^{3}$ .

$x (5 x^{2}) - 5 x^{2} - 12 x = 0$

Step 2.2

Factor $x$ out of $- 5 x^{2}$ .

$x (5 x^{2}) + x (- 5 x) - 12 x = 0$

Step 2.3

Factor $x$ out of $- 12 x$ .

$x (5 x^{2}) + x (- 5 x) + x \cdot - 12 = 0$

Step 2.4

Factor $x$ out of $x (5 x^{2}) + x (- 5 x)$ .

$x (5 x^{2} - 5 x) + x \cdot - 12 = 0$

Step 2.5

Factor $x$ out of $x (5 x^{2} - 5 x) + x \cdot - 12$ .

$x (5 x^{2} - 5 x - 12) = 0$

Step 3

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

$x = 0$

$5 x^{2} - 5 x - 12 = 0$

Step 4

Set $x$ equal to $0$ .

$x = 0$

Step 5

Set $5 x^{2} - 5 x - 12$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $5 x^{2} - 5 x - 12$ equal to $0$ .

$5 x^{2} - 5 x - 12 = 0$

Step 5.2

Solve $5 x^{2} - 5 x - 12 = 0$ for $x$ .

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Step 5.2.1

Use the quadratic formula to find the solutions.

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

Step 5.2.2

Substitute the values $a = 5$ , $b = - 5$ , and $c = - 12$ into the quadratic formula and solve for $x$ .

$\frac{5 \pm \sqrt{{(- 5)}^{2} - 4 \cdot (5 \cdot - 12)}}{2 \cdot 5}$

Step 5.2.3

Simplify.

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Step 5.2.3.1

Simplify the numerator.

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Step 5.2.3.1.1

Raise $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 5 \cdot - 12}}{2 \cdot 5}$

Step 5.2.3.1.2

Multiply $- 4 \cdot 5 \cdot - 12$ .

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Step 5.2.3.1.2.1

Multiply $- 4$ by $5$ .

$x = \frac{5 \pm \sqrt{25 - 20 \cdot - 12}}{2 \cdot 5}$

Step 5.2.3.1.2.2

Multiply $- 20$ by $- 12$ .

$x = \frac{5 \pm \sqrt{25 + 240}}{2 \cdot 5}$

Step 5.2.3.1.3

Add $25$ and $240$ .

$x = \frac{5 \pm \sqrt{265}}{2 \cdot 5}$

Step 5.2.3.2

Multiply $2$ by $5$ .

$x = \frac{5 \pm \sqrt{265}}{10}$

Step 5.2.4

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

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Step 5.2.4.1

Simplify the numerator.

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Step 5.2.4.1.1

Raise $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 5 \cdot - 12}}{2 \cdot 5}$

Step 5.2.4.1.2

Multiply $- 4 \cdot 5 \cdot - 12$ .

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Step 5.2.4.1.2.1

Multiply $- 4$ by $5$ .

$x = \frac{5 \pm \sqrt{25 - 20 \cdot - 12}}{2 \cdot 5}$

Step 5.2.4.1.2.2

Multiply $- 20$ by $- 12$ .

$x = \frac{5 \pm \sqrt{25 + 240}}{2 \cdot 5}$

Step 5.2.4.1.3

Add $25$ and $240$ .

$x = \frac{5 \pm \sqrt{265}}{2 \cdot 5}$

Step 5.2.4.2

Multiply $2$ by $5$ .

$x = \frac{5 \pm \sqrt{265}}{10}$

Step 5.2.4.3

Change the $\pm$ to $+$ .

$x = \frac{5 + \sqrt{265}}{10}$

Step 5.2.5

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

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Step 5.2.5.1

Simplify the numerator.

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Step 5.2.5.1.1

Raise $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 5 \cdot - 12}}{2 \cdot 5}$

Step 5.2.5.1.2

Multiply $- 4 \cdot 5 \cdot - 12$ .

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Step 5.2.5.1.2.1

Multiply $- 4$ by $5$ .

$x = \frac{5 \pm \sqrt{25 - 20 \cdot - 12}}{2 \cdot 5}$

Step 5.2.5.1.2.2

Multiply $- 20$ by $- 12$ .

$x = \frac{5 \pm \sqrt{25 + 240}}{2 \cdot 5}$

Step 5.2.5.1.3

Add $25$ and $240$ .

$x = \frac{5 \pm \sqrt{265}}{2 \cdot 5}$

Step 5.2.5.2

Multiply $2$ by $5$ .

$x = \frac{5 \pm \sqrt{265}}{10}$

Step 5.2.5.3

Change the $\pm$ to $-$ .

$x = \frac{5 - \sqrt{265}}{10}$

Step 5.2.6

The final answer is the combination of both solutions.

$x = \frac{5 + \sqrt{265}}{10}, \frac{5 - \sqrt{265}}{10}$

Step 6

The final solution is all the values that make $x (5 x^{2} - 5 x - 12) = 0$ true.

$x = 0, \frac{5 + \sqrt{265}}{10}, \frac{5 - \sqrt{265}}{10}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = 0, \frac{5 + \sqrt{265}}{10}, \frac{5 - \sqrt{265}}{10}$

Decimal Form:

$x = 0, 2.12788205 \dots, - 1.12788205 \dots$