Algebra Examples

$9 x^{3} - 3 x^{2} - 3 x + 1 = 0$

Step 1

Factor out the greatest common factor from each group.

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Step 1.1

Group the first two terms and the last two terms.

$(9 x^{3} - 3 x^{2}) - 3 x + 1 = 0$

Step 1.2

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

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

Step 2

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

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

Step 3

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

$3 x - 1 = 0$

$3 x^{2} - 1 = 0$

Step 4

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

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Step 4.1

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

$3 x - 1 = 0$

Step 4.2

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

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Step 4.2.1

Add $1$ to both sides of the equation.

$3 x = 1$

Step 4.2.2

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

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Step 4.2.2.1

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

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

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Cancel the common factor of $3$ .

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Step 4.2.2.2.1.1

Cancel the common factor.

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

Step 4.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 5

Set $3 x^{2} - 1$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $3 x^{2} - 1$ equal to $0$ .

$3 x^{2} - 1 = 0$

Step 5.2

Solve $3 x^{2} - 1 = 0$ for $x$ .

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Step 5.2.1

Add $1$ to both sides of the equation.

$3 x^{2} = 1$

Step 5.2.2

Divide each term in $3 x^{2} = 1$ by $3$ and simplify.

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Step 5.2.2.1

Divide each term in $3 x^{2} = 1$ by $3$ .

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

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of $3$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

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

Step 5.2.2.2.1.2

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

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

Step 5.2.3

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

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

Step 5.2.4

Simplify $\pm \sqrt{\frac{1}{3}}$ .

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Step 5.2.4.1

Rewrite $\sqrt{\frac{1}{3}}$ as $\frac{\sqrt{1}}{\sqrt{3}}$ .

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

Step 5.2.4.2

Any root of $1$ is $1$ .

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

Step 5.2.4.3

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 5.2.4.4

Combine and simplify the denominator.

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Step 5.2.4.4.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 5.2.4.4.2

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

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

Step 5.2.4.4.3

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

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

Step 5.2.4.4.4

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

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

Step 5.2.4.4.5

Add $1$ and $1$ .

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

Step 5.2.4.4.6

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

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Step 5.2.4.4.6.1

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

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

Step 5.2.4.4.6.2

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

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

Step 5.2.4.4.6.3

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

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

Step 5.2.4.4.6.4

Cancel the common factor of $2$ .

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Step 5.2.4.4.6.4.1

Cancel the common factor.

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

Step 5.2.4.4.6.4.2

Rewrite the expression.

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

Step 5.2.4.4.6.5

Evaluate the exponent.

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

Step 5.2.5

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

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Step 5.2.5.1

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

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

Step 5.2.5.2

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

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

Step 5.2.5.3

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

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

Step 6

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

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.\overline{3}, 0.57735026 \dots, - 0.57735026 \dots$