Precalculus Examples

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

Step 1

Add $1$ to both sides of the equation.

$9 x^{2} = 1$

Step 2

Divide each term in $9 x^{2} = 1$ by $9$ and simplify.

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Step 2.1

Divide each term in $9 x^{2} = 1$ by $9$ .

$\frac{9 x^{2}}{9} = \frac{1}{9}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $9$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{9 x^{2}}{9} = \frac{1}{9}$

Step 2.2.1.2

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

$x^{2} = \frac{1}{9}$

Step 3

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

$x = \pm \sqrt{\frac{1}{9}}$

Step 4

Simplify $\pm \sqrt{\frac{1}{9}}$ .

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Step 4.1

Rewrite $\sqrt{\frac{1}{9}}$ as $\frac{\sqrt{1}}{\sqrt{9}}$ .

$x = \pm \frac{\sqrt{1}}{\sqrt{9}}$

Step 4.2

Any root of $1$ is $1$ .

$x = \pm \frac{1}{\sqrt{9}}$

Step 4.3

Simplify the denominator.

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Step 4.3.1

Rewrite $9$ as $3^{2}$ .

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

Step 4.3.2

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

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

Step 5

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

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

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