Algebra Examples

$12 x^{4} - 48 x^{3} - 8 x^{2} + 32 x$

Step 1

Set $12 x^{4} - 48 x^{3} - 8 x^{2} + 32 x$ equal to $0$ .

$12 x^{4} - 48 x^{3} - 8 x^{2} + 32 x = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor the left side of the equation.

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Step 2.1.1

Factor $4 x$ out of $12 x^{4} - 48 x^{3} - 8 x^{2} + 32 x$ .

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Step 2.1.1.1

Factor $4 x$ out of $12 x^{4}$ .

$4 x (3 x^{3}) - 48 x^{3} - 8 x^{2} + 32 x = 0$

Step 2.1.1.2

Factor $4 x$ out of $- 48 x^{3}$ .

$4 x (3 x^{3}) + 4 x (- 12 x^{2}) - 8 x^{2} + 32 x = 0$

Step 2.1.1.3

Factor $4 x$ out of $- 8 x^{2}$ .

$4 x (3 x^{3}) + 4 x (- 12 x^{2}) + 4 x (- 2 x) + 32 x = 0$

Step 2.1.1.4

Factor $4 x$ out of $32 x$ .

$4 x (3 x^{3}) + 4 x (- 12 x^{2}) + 4 x (- 2 x) + 4 x (8) = 0$

Step 2.1.1.5

Factor $4 x$ out of $4 x (3 x^{3}) + 4 x (- 12 x^{2})$ .

$4 x (3 x^{3} - 12 x^{2}) + 4 x (- 2 x) + 4 x (8) = 0$

Step 2.1.1.6

Factor $4 x$ out of $4 x (3 x^{3} - 12 x^{2}) + 4 x (- 2 x)$ .

$4 x (3 x^{3} - 12 x^{2} - 2 x) + 4 x (8) = 0$

Step 2.1.1.7

Factor $4 x$ out of $4 x (3 x^{3} - 12 x^{2} - 2 x) + 4 x (8)$ .

$4 x (3 x^{3} - 12 x^{2} - 2 x + 8) = 0$

Step 2.1.2

Factor out the greatest common factor from each group.

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Step 2.1.2.1

Group the first two terms and the last two terms.

$4 x ((3 x^{3} - 12 x^{2}) - 2 x + 8) = 0$

Step 2.1.2.2

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

$4 x (3 x^{2} (x - 4) - 2 (x - 4)) = 0$

Step 2.1.3

Factor.

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Step 2.1.3.1

Factor the polynomial by factoring out the greatest common factor, $x - 4$ .

$4 x ((x - 4) (3 x^{2} - 2)) = 0$

Step 2.1.3.2

Remove unnecessary parentheses.

$4 x (x - 4) (3 x^{2} - 2) = 0$

Step 2.2

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

$x = 0$

$x - 4 = 0$

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

Step 2.3

Set $x$ equal to $0$ .

$x = 0$

Step 2.4

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 2.4.2

Add $4$ to both sides of the equation.

$x = 4$

Step 2.5

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

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Step 2.5.1

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

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

Step 2.5.2

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

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Step 2.5.2.1

Add $2$ to both sides of the equation.

$3 x^{2} = 2$

Step 2.5.2.2

Divide each term in $3 x^{2} = 2$ by $3$ and simplify.

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Step 2.5.2.2.1

Divide each term in $3 x^{2} = 2$ by $3$ .

$\frac{3 x^{2}}{3} = \frac{2}{3}$

Step 2.5.2.2.2

Simplify the left side.

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Step 2.5.2.2.2.1

Cancel the common factor of $3$ .

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Step 2.5.2.2.2.1.1

Cancel the common factor.

$\frac{3 x^{2}}{3} = \frac{2}{3}$

Step 2.5.2.2.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{2}{3}$

Step 2.5.2.3

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

$x = \pm \sqrt{\frac{2}{3}}$

Step 2.5.2.4

Simplify $\pm \sqrt{\frac{2}{3}}$ .

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Step 2.5.2.4.1

Rewrite $\sqrt{\frac{2}{3}}$ as $\frac{\sqrt{2}}{\sqrt{3}}$ .

$x = \pm \frac{\sqrt{2}}{\sqrt{3}}$

Step 2.5.2.4.2

Multiply $\frac{\sqrt{2}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm \frac{\sqrt{2}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 2.5.2.4.3

Combine and simplify the denominator.

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Step 2.5.2.4.3.1

Multiply $\frac{\sqrt{2}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm \frac{\sqrt{2} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 2.5.2.4.3.2

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm \frac{\sqrt{2} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 2.5.2.4.3.3

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm \frac{\sqrt{2} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 2.5.2.4.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \pm \frac{\sqrt{2} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 2.5.2.4.3.5

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{2} \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 2.5.2.4.3.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.5.2.4.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$x = \pm \frac{\sqrt{2} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 2.5.2.4.3.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \pm \frac{\sqrt{2} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 2.5.2.4.3.6.3

Combine $\frac{1}{2}$ and $2$ .

$x = \pm \frac{\sqrt{2} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 2.5.2.4.3.6.4

Cancel the common factor of $2$ .

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Step 2.5.2.4.3.6.4.1

Cancel the common factor.

$x = \pm \frac{\sqrt{2} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 2.5.2.4.3.6.4.2

Rewrite the expression.

$x = \pm \frac{\sqrt{2} \sqrt{3}}{3^{1}}$

Step 2.5.2.4.3.6.5

Evaluate the exponent.

$x = \pm \frac{\sqrt{2} \sqrt{3}}{3}$

Step 2.5.2.4.4

Simplify the numerator.

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Step 2.5.2.4.4.1

Combine using the product rule for radicals.

$x = \pm \frac{\sqrt{2 \cdot 3}}{3}$

Step 2.5.2.4.4.2

Multiply $2$ by $3$ .

$x = \pm \frac{\sqrt{6}}{3}$

Step 2.5.2.5

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

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Step 2.5.2.5.1

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

$x = \frac{\sqrt{6}}{3}$

Step 2.5.2.5.2

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

$x = - \frac{\sqrt{6}}{3}$

Step 2.5.2.5.3

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

$x = \frac{\sqrt{6}}{3}, - \frac{\sqrt{6}}{3}$

Step 2.6

The final solution is all the values that make $4 x (x - 4) (3 x^{2} - 2) = 0$ true.

$x = 0, 4, \frac{\sqrt{6}}{3}, - \frac{\sqrt{6}}{3}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = 0, 4, \frac{\sqrt{6}}{3}, - \frac{\sqrt{6}}{3}$

Decimal Form:

$x = 0, 4, 0.81649658 \dots, - 0.81649658 \dots$

Step 4