Algebra Examples

$\frac{4}{25 - 4 x^{2}}$

Step 1

Set the denominator in $\frac{4}{25 - 4 x^{2}}$ equal to $0$ to find where the expression is undefined.

$25 - 4 x^{2} = 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $25$ from both sides of the equation.

$- 4 x^{2} = - 25$

Step 2.2

Divide each term in $- 4 x^{2} = - 25$ by $- 4$ and simplify.

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Step 2.2.1

Divide each term in $- 4 x^{2} = - 25$ by $- 4$ .

$\frac{- 4 x^{2}}{- 4} = \frac{- 25}{- 4}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $- 4$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{- 4 x^{2}}{- 4} = \frac{- 25}{- 4}$

Step 2.2.2.1.2

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

$x^{2} = \frac{- 25}{- 4}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Dividing two negative values results in a positive value.

$x^{2} = \frac{25}{4}$

Step 2.3

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

$x = \pm \sqrt{\frac{25}{4}}$

Step 2.4

Simplify $\pm \sqrt{\frac{25}{4}}$ .

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Step 2.4.1

Rewrite $\sqrt{\frac{25}{4}}$ as $\frac{\sqrt{25}}{\sqrt{4}}$ .

$x = \pm \frac{\sqrt{25}}{\sqrt{4}}$

Step 2.4.2

Simplify the numerator.

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Step 2.4.2.1

Rewrite $25$ as $5^{2}$ .

$x = \pm \frac{\sqrt{5^{2}}}{\sqrt{4}}$

Step 2.4.2.2

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

$x = \pm \frac{5}{\sqrt{4}}$

Step 2.4.3

Simplify the denominator.

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Step 2.4.3.1

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

$x = \pm \frac{5}{\sqrt{2^{2}}}$

Step 2.4.3.2

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

$x = \pm \frac{5}{2}$

Step 2.5

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

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Step 2.5.1

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

$x = \frac{5}{2}$

Step 2.5.2

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

$x = - \frac{5}{2}$

Step 2.5.3

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

$x = \frac{5}{2}, - \frac{5}{2}$

Step 3

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

Interval Notation:

$(- \infty, - \frac{5}{2}) \cup (- \frac{5}{2}, \frac{5}{2}) \cup (\frac{5}{2}, \infty)$

Set-Builder Notation:

${x | x \neq \frac{5}{2}, - \frac{5}{2}}$

Step 4