Solve by Factoring -19x^3-12x^2+16x=0

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

Step 1

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

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

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

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

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

Factor $x$ out of $16 x$ .

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

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

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

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

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

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 $x$ .

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

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

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

Use the quadratic formula to find the solutions.

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

Substitute the values $a = - 19$ , $b = - 12$ , and $c = 16$ into the quadratic formula and solve for $x$ .

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

Simplify.

Simplify the numerator.

Raise $- 12$ to the power of $2$ .

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

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

Multiply $- 4$ by $- 19$ .

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

Multiply $76$ by $16$ .

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

Add $144$ and $1216$ .

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

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

Factor $16$ out of $1360$ .

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

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

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

Pull terms out from under the radical.

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

Multiply $2$ by $- 19$ .

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

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

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

Move the negative in front of the fraction.

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

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

Simplify the numerator.

Raise $- 12$ to the power of $2$ .

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

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

Multiply $- 4$ by $- 19$ .

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

Multiply $76$ by $16$ .

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

Add $144$ and $1216$ .

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

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

Factor $16$ out of $1360$ .

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

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

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

Pull terms out from under the radical.

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

Multiply $2$ by $- 19$ .

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

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

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

Move the negative in front of the fraction.

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

Change the $\pm$ to $+$ .

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

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

Simplify the numerator.

Raise $- 12$ to the power of $2$ .

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

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

Multiply $- 4$ by $- 19$ .

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

Multiply $76$ by $16$ .

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

Add $144$ and $1216$ .

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

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

Factor $16$ out of $1360$ .

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

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

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

Pull terms out from under the radical.

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

Multiply $2$ by $- 19$ .

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

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

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

Move the negative in front of the fraction.

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

Change the $\pm$ to $-$ .

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

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:

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

Solve by Factoring -19x^3-12x^2+16x=0

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