Algebra Examples

$y = \frac{1}{\sqrt{x^{2}} - 4}$

Step 1

Set the radicand in $\sqrt{x^{2}}$ greater than or equal to $0$ to find where the expression is defined.

$x^{2} \geq 0$

Step 2

Since the left side has an even power, it is always positive for all real numbers.

All real numbers

Step 3

Set the denominator in $\frac{1}{\sqrt{x^{2}} - 4}$ equal to $0$ to find where the expression is undefined.

$\sqrt{x^{2}} - 4 = 0$

Step 4

Solve for $x$ .

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Step 4.1

Add $4$ to both sides of the equation.

$\sqrt{x^{2}} = 4$

Step 4.2

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{x^{2}}}^{2} = 4^{2}$

Step 4.3

Simplify each side of the equation.

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Step 4.3.1

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

${(x^{\frac{2}{2}})}^{2} = 4^{2}$

Step 4.3.2

Divide $2$ by $2$ .

${(x^{1})}^{2} = 4^{2}$

Step 4.3.3

Simplify the left side.

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Step 4.3.3.1

Multiply the exponents in ${(x^{1})}^{2}$ .

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Step 4.3.3.1.1

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

$x^{1 \cdot 2} = 4^{2}$

Step 4.3.3.1.2

Multiply $2$ by $1$ .

$x^{2} = 4^{2}$

Step 4.3.4

Simplify the right side.

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Step 4.3.4.1

Raise $4$ to the power of $2$ .

$x^{2} = 16$

Step 4.4

Solve for $x$ .

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Step 4.4.1

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

$x = \pm \sqrt{16}$

Step 4.4.2

Simplify $\pm \sqrt{16}$ .

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Step 4.4.2.1

Rewrite $16$ as $4^{2}$ .

$x = \pm \sqrt{4^{2}}$

Step 4.4.2.2

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

$x = \pm 4$

Step 4.4.3

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

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Step 4.4.3.1

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

$x = 4$

Step 4.4.3.2

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

$x = - 4$

Step 4.4.3.3

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

$x = 4, - 4$

Step 5

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, - 4) \cup (- 4, 4) \cup (4, \infty)$

Set-Builder Notation:

${x | x \neq 4, - 4}$

Step 6