Algebra Examples

$- 19 x^{3} - 12 x^{2} + 16 x = 0$

Step 1

Factor $- x$ out of $- 19 x^{3} - 12 x^{2} + 16 x$ .

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Step 1.1

Factor $- x$ out of $- 19 x^{3}$ .

$- x (19 x^{2}) - 12 x^{2} + 16 x = 0$

Step 1.2

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

$- x (19 x^{2}) - x (12 x) + 16 x = 0$

Step 1.3

Factor $- x$ out of $16 x$ .

$- x (19 x^{2}) - x (12 x) - x \cdot - 16 = 0$

Step 1.4

Factor $- x$ out of $- x (19 x^{2}) - x (12 x)$ .

$- x (19 x^{2} + 12 x) - x \cdot - 16 = 0$

Step 1.5

Factor $- x$ out of $- x (19 x^{2} + 12 x) - x (- 16)$ .

$- x (19 x^{2} + 12 x - 16) = 0$

Step 2

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

$x = 0$

$19 x^{2} + 12 x - 16 = 0$

Step 3

Set $x$ equal to $0$ .

$x = 0$

Step 4

Set $19 x^{2} + 12 x - 16$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $19 x^{2} + 12 x - 16$ equal to $0$ .

$19 x^{2} + 12 x - 16 = 0$

Step 4.2

Solve $19 x^{2} + 12 x - 16 = 0$ 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 $a = 19$ , $b = 12$ , and $c = - 16$ into the quadratic formula and solve for $x$ .

$\frac{- 12 \pm \sqrt{12^{2} - 4 \cdot (19 \cdot - 16)}}{2 \cdot 19}$

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 $12$ to the power of $2$ .

$x = \frac{- 12 \pm \sqrt{144 - 4 \cdot 19 \cdot - 16}}{2 \cdot 19}$

Step 4.2.3.1.2

Multiply $- 4 \cdot 19 \cdot - 16$ .

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Step 4.2.3.1.2.1

Multiply $- 4$ by $19$ .

$x = \frac{- 12 \pm \sqrt{144 - 76 \cdot - 16}}{2 \cdot 19}$

Step 4.2.3.1.2.2

Multiply $- 76$ by $- 16$ .

$x = \frac{- 12 \pm \sqrt{144 + 1216}}{2 \cdot 19}$

Step 4.2.3.1.3

Add $144$ and $1216$ .

$x = \frac{- 12 \pm \sqrt{1360}}{2 \cdot 19}$

Step 4.2.3.1.4

Rewrite $1360$ as $4^{2} \cdot 85$ .

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Step 4.2.3.1.4.1

Factor $16$ out of $1360$ .

$x = \frac{- 12 \pm \sqrt{16 (85)}}{2 \cdot 19}$

Step 4.2.3.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{- 12 \pm \sqrt{4^{2} \cdot 85}}{2 \cdot 19}$

Step 4.2.3.1.5

Pull terms out from under the radical.

$x = \frac{- 12 \pm 4 \sqrt{85}}{2 \cdot 19}$

Step 4.2.3.2

Multiply $2$ by $19$ .

$x = \frac{- 12 \pm 4 \sqrt{85}}{38}$

Step 4.2.3.3

Simplify $\frac{- 12 \pm 4 \sqrt{85}}{38}$ .

$x = \frac{- 6 \pm 2 \sqrt{85}}{19}$

Step 4.2.4

The final answer is the combination of both solutions.

$x = - \frac{6 - 2 \sqrt{85}}{19}, - \frac{6 + 2 \sqrt{85}}{19}$

Step 5

The final solution is all the values that make $- x (19 x^{2} + 12 x - 16) = 0$ true.

$x = 0, - \frac{6 - 2 \sqrt{85}}{19}, - \frac{6 + 2 \sqrt{85}}{19}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = 0, - \frac{6 - 2 \sqrt{85}}{19}, - \frac{6 + 2 \sqrt{85}}{19}$

Decimal Form:

$x = 0, 0.65468889 \dots, - 1.28626783 \dots$