Precalculus Examples

$3 x^{3} - x^{2} - 36 x + 12$

Step 1

Factor the left side of the equation.

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Step 1.1

Factor out the greatest common factor from each group.

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Step 1.1.1

Group the first two terms and the last two terms.

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

Step 1.1.2

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

$x^{2} (3 x - 1) - 12 (3 x - 1) = 0$

Step 1.2

Factor the polynomial by factoring out the greatest common factor, $3 x - 1$ .

$(3 x - 1) (x^{2} - 12) = 0$

Step 2

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

$3 x - 1 = 0$

$x^{2} - 12 = 0$

Step 3

Set $3 x - 1$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $3 x - 1$ equal to $0$ .

$3 x - 1 = 0$

Step 3.2

Solve $3 x - 1 = 0$ for $x$ .

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Step 3.2.1

Add $1$ to both sides of the equation.

$3 x = 1$

Step 3.2.2

Divide each term in $3 x = 1$ by $3$ and simplify.

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Step 3.2.2.1

Divide each term in $3 x = 1$ by $3$ .

$\frac{3 x}{3} = \frac{1}{3}$

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Cancel the common factor of $3$ .

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Step 3.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{1}{3}$

Step 3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{3}$

Step 4

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

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Step 4.1

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

$x^{2} - 12 = 0$

Step 4.2

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

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Step 4.2.1

Add $12$ to both sides of the equation.

$x^{2} = 12$

Step 4.2.2

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

$x = \pm \sqrt{12}$

Step 4.2.3

Simplify $\pm \sqrt{12}$ .

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Step 4.2.3.1

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 4.2.3.1.1

Factor $4$ out of $12$ .

$x = \pm \sqrt{4 (3)}$

Step 4.2.3.1.2

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

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

Step 4.2.3.2

Pull terms out from under the radical.

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

Step 4.2.4

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

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Step 4.2.4.1

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

$x = 2 \sqrt{3}$

Step 4.2.4.2

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

$x = - 2 \sqrt{3}$

Step 4.2.4.3

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

$x = 2 \sqrt{3}, - 2 \sqrt{3}$

Step 5

The final solution is all the values that make $(3 x - 1) (x^{2} - 12) = 0$ true.

$x = \frac{1}{3}, 2 \sqrt{3}, - 2 \sqrt{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{3}, 2 \sqrt{3}, - 2 \sqrt{3}$

Decimal Form:

$x = 0.\overline{3}, 3.46410161 \dots, - 3.46410161 \dots$

Step 7