Algebra Examples

$18 x^{3} - 15 x^{2} - 9 x = 0$

Step 1

Factor $3 x$ out of $18 x^{3} - 15 x^{2} - 9 x$ .

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Step 1.1

Factor $3 x$ out of $18 x^{3}$ .

$3 x (6 x^{2}) - 15 x^{2} - 9 x = 0$

Step 1.2

Factor $3 x$ out of $- 15 x^{2}$ .

$3 x (6 x^{2}) + 3 x (- 5 x) - 9 x = 0$

Step 1.3

Factor $3 x$ out of $- 9 x$ .

$3 x (6 x^{2}) + 3 x (- 5 x) + 3 x (- 3) = 0$

Step 1.4

Factor $3 x$ out of $3 x (6 x^{2}) + 3 x (- 5 x)$ .

$3 x (6 x^{2} - 5 x) + 3 x (- 3) = 0$

Step 1.5

Factor $3 x$ out of $3 x (6 x^{2} - 5 x) + 3 x (- 3)$ .

$3 x (6 x^{2} - 5 x - 3) = 0$

Step 2

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

$x = 0$

$6 x^{2} - 5 x - 3 = 0$

Step 3

Set $x$ equal to $0$ .

$x = 0$

Step 4

Set $6 x^{2} - 5 x - 3$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $6 x^{2} - 5 x - 3$ equal to $0$ .

$6 x^{2} - 5 x - 3 = 0$

Step 4.2

Solve $6 x^{2} - 5 x - 3 = 0$ for $x$ .

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Step 4.2.1

Use the quadratic formula to find the solutions.

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

Step 4.2.2

Substitute the values $a = 6$ , $b = - 5$ , and $c = - 3$ into the quadratic formula and solve for $x$ .

$\frac{5 \pm \sqrt{{(- 5)}^{2} - 4 \cdot (6 \cdot - 3)}}{2 \cdot 6}$

Step 4.2.3

Simplify.

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Step 4.2.3.1

Simplify the numerator.

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Step 4.2.3.1.1

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

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 6 \cdot - 3}}{2 \cdot 6}$

Step 4.2.3.1.2

Multiply $- 4 \cdot 6 \cdot - 3$ .

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Step 4.2.3.1.2.1

Multiply $- 4$ by $6$ .

$x = \frac{5 \pm \sqrt{25 - 24 \cdot - 3}}{2 \cdot 6}$

Step 4.2.3.1.2.2

Multiply $- 24$ by $- 3$ .

$x = \frac{5 \pm \sqrt{25 + 72}}{2 \cdot 6}$

Step 4.2.3.1.3

Add $25$ and $72$ .

$x = \frac{5 \pm \sqrt{97}}{2 \cdot 6}$

Step 4.2.3.2

Multiply $2$ by $6$ .

$x = \frac{5 \pm \sqrt{97}}{12}$

Step 4.2.4

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

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Step 4.2.4.1

Simplify the numerator.

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Step 4.2.4.1.1

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

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 6 \cdot - 3}}{2 \cdot 6}$

Step 4.2.4.1.2

Multiply $- 4 \cdot 6 \cdot - 3$ .

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Step 4.2.4.1.2.1

Multiply $- 4$ by $6$ .

$x = \frac{5 \pm \sqrt{25 - 24 \cdot - 3}}{2 \cdot 6}$

Step 4.2.4.1.2.2

Multiply $- 24$ by $- 3$ .

$x = \frac{5 \pm \sqrt{25 + 72}}{2 \cdot 6}$

Step 4.2.4.1.3

Add $25$ and $72$ .

$x = \frac{5 \pm \sqrt{97}}{2 \cdot 6}$

Step 4.2.4.2

Multiply $2$ by $6$ .

$x = \frac{5 \pm \sqrt{97}}{12}$

Step 4.2.4.3

Change the $\pm$ to $+$ .

$x = \frac{5 + \sqrt{97}}{12}$

Step 4.2.5

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

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Step 4.2.5.1

Simplify the numerator.

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Step 4.2.5.1.1

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

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 6 \cdot - 3}}{2 \cdot 6}$

Step 4.2.5.1.2

Multiply $- 4 \cdot 6 \cdot - 3$ .

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Step 4.2.5.1.2.1

Multiply $- 4$ by $6$ .

$x = \frac{5 \pm \sqrt{25 - 24 \cdot - 3}}{2 \cdot 6}$

Step 4.2.5.1.2.2

Multiply $- 24$ by $- 3$ .

$x = \frac{5 \pm \sqrt{25 + 72}}{2 \cdot 6}$

Step 4.2.5.1.3

Add $25$ and $72$ .

$x = \frac{5 \pm \sqrt{97}}{2 \cdot 6}$

Step 4.2.5.2

Multiply $2$ by $6$ .

$x = \frac{5 \pm \sqrt{97}}{12}$

Step 4.2.5.3

Change the $\pm$ to $-$ .

$x = \frac{5 - \sqrt{97}}{12}$

Step 4.2.6

The final answer is the combination of both solutions.

$x = \frac{5 + \sqrt{97}}{12}, \frac{5 - \sqrt{97}}{12}$

Step 5

The final solution is all the values that make $3 x (6 x^{2} - 5 x - 3) = 0$ true.

$x = 0, \frac{5 + \sqrt{97}}{12}, \frac{5 - \sqrt{97}}{12}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = 0, \frac{5 + \sqrt{97}}{12}, \frac{5 - \sqrt{97}}{12}$

Decimal Form:

$x = 0, 1.23740481 \dots, - 0.40407148 \dots$