Solve for x 12x^6+2x^5-72x^4-12x^3=0

$12 x^{6} + 2 x^{5} - 72 x^{4} - 12 x^{3} = 0$

Factor the left side of the equation.

Factor $2 x^{3}$ out of $12 x^{6} + 2 x^{5} - 72 x^{4} - 12 x^{3}$ .

Factor $2 x^{3}$ out of $12 x^{6}$ .

$2 x^{3} (6 x^{3}) + 2 x^{5} - 72 x^{4} - 12 x^{3} = 0$

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

$2 x^{3} (6 x^{3}) + 2 x^{3} (x^{2}) - 72 x^{4} - 12 x^{3} = 0$

Factor $2 x^{3}$ out of $- 72 x^{4}$ .

$2 x^{3} (6 x^{3}) + 2 x^{3} (x^{2}) + 2 x^{3} (- 36 x) - 12 x^{3} = 0$

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

$2 x^{3} (6 x^{3}) + 2 x^{3} (x^{2}) + 2 x^{3} (- 36 x) + 2 x^{3} (- 6) = 0$

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

$2 x^{3} (6 x^{3} + x^{2}) + 2 x^{3} (- 36 x) + 2 x^{3} (- 6) = 0$

Factor $2 x^{3}$ out of $2 x^{3} (6 x^{3} + x^{2}) + 2 x^{3} (- 36 x)$ .

$2 x^{3} (6 x^{3} + x^{2} - 36 x) + 2 x^{3} (- 6) = 0$

Factor $2 x^{3}$ out of $2 x^{3} (6 x^{3} + x^{2} - 36 x) + 2 x^{3} (- 6)$ .

$2 x^{3} (6 x^{3} + x^{2} - 36 x - 6) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$2 x^{3} ((6 x^{3} + x^{2}) - 36 x - 6) = 0$

Factor out the greatest common factor (GCF) from each group.

$2 x^{3} (x^{2} (6 x + 1) - 6 (6 x + 1)) = 0$

Factor.

Factor the polynomial by factoring out the greatest common factor, $6 x + 1$ .

$2 x^{3} ((6 x + 1) (x^{2} - 6)) = 0$

Remove unnecessary parentheses.

$2 x^{3} (6 x + 1) (x^{2} - 6) = 0$

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

$x^{3} = 0$

$6 x + 1 = 0$

$x^{2} - 6 = 0$

Set $x^{3}$ equal to $0$ and solve for $x$ .

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

$x^{3} = 0$

Solve $x^{3} = 0$ for $x$ .

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

$x = \sqrt[3]{0}$

Simplify $\sqrt[3]{0}$ .

Rewrite $0$ as $0^{3}$ .

$x = \sqrt[3]{0^{3}}$

Pull terms out from under the radical, assuming real numbers.

$x = 0$

Set $6 x + 1$ equal to $0$ and solve for $x$ .

Set $6 x + 1$ equal to $0$ .

$6 x + 1 = 0$

Solve $6 x + 1 = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$6 x = - 1$

Divide each term in $6 x = - 1$ by $6$ and simplify.

Divide each term in $6 x = - 1$ by $6$ .

$\frac{6 x}{6} = \frac{- 1}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 x}{6} = \frac{- 1}{6}$

Divide $x$ by $1$ .

$x = \frac{- 1}{6}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{1}{6}$

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

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

$x^{2} - 6 = 0$

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

Add $6$ to both sides of the equation.

$x^{2} = 6$

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

$x = \pm \sqrt{6}$

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

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

$x = \sqrt{6}$

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

$x = - \sqrt{6}$

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

$x = \sqrt{6}, - \sqrt{6}$

The final solution is all the values that make $2 x^{3} (6 x + 1) (x^{2} - 6) = 0$ true.

$x = 0, - \frac{1}{6}, \sqrt{6}, - \sqrt{6}$

The result can be shown in multiple forms.

Exact Form:

$x = 0, - \frac{1}{6}, \sqrt{6}, - \sqrt{6}$

Decimal Form:

$x = 0, - 0.1\overline{6}, 2.44948974 \dots, - 2.44948974 \dots$

Solve for x 12x^6+2x^5-72x^4-12x^3=0

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