Linear Algebra Examples

$\frac{- 14 x}{6 x^{2} - x^{2} - 2}$

Step 1

Set the denominator in $\frac{- 14 x}{6 x^{2} - x^{2} - 2}$ equal to $0$ to find where the expression is undefined.

$6 x^{2} - x^{2} - 2 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $x^{2}$ from $6 x^{2}$ .

$5 x^{2} - 2 = 0$

Step 2.2

Add $2$ to both sides of the equation.

$5 x^{2} = 2$

Step 2.3

Divide each term in $5 x^{2} = 2$ by $5$ and simplify.

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Step 2.3.1

Divide each term in $5 x^{2} = 2$ by $5$ .

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

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $5$ .

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Step 2.3.2.1.1

Cancel the common factor.

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

Step 2.3.2.1.2

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

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

Step 2.4

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

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

Step 2.5

Simplify $\pm \sqrt{\frac{2}{5}}$ .

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Step 2.5.1

Rewrite $\sqrt{\frac{2}{5}}$ as $\frac{\sqrt{2}}{\sqrt{5}}$ .

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

Step 2.5.2

Multiply $\frac{\sqrt{2}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

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

Step 2.5.3

Combine and simplify the denominator.

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Step 2.5.3.1

Multiply $\frac{\sqrt{2}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

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

Step 2.5.3.2

Raise $\sqrt{5}$ to the power of $1$ .

$x = \pm \frac{\sqrt{2} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Step 2.5.3.3

Raise $\sqrt{5}$ to the power of $1$ .

$x = \pm \frac{\sqrt{2} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

Step 2.5.3.4

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

$x = \pm \frac{\sqrt{2} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 2.5.3.5

Add $1$ and $1$ .

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

Step 2.5.3.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 2.5.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$x = \pm \frac{\sqrt{2} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 2.5.3.6.2

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

$x = \pm \frac{\sqrt{2} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 2.5.3.6.3

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

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

Step 2.5.3.6.4

Cancel the common factor of $2$ .

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Step 2.5.3.6.4.1

Cancel the common factor.

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

Step 2.5.3.6.4.2

Rewrite the expression.

$x = \pm \frac{\sqrt{2} \sqrt{5}}{5^{1}}$

Step 2.5.3.6.5

Evaluate the exponent.

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

Step 2.5.4

Simplify the numerator.

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Step 2.5.4.1

Combine using the product rule for radicals.

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

Step 2.5.4.2

Multiply $2$ by $5$ .

$x = \pm \frac{\sqrt{10}}{5}$

Step 2.6

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

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Step 2.6.1

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

$x = \frac{\sqrt{10}}{5}$

Step 2.6.2

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

$x = - \frac{\sqrt{10}}{5}$

Step 2.6.3

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

$x = \frac{\sqrt{10}}{5}, - \frac{\sqrt{10}}{5}$

Step 3

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

Interval Notation:

$(- \infty, - \frac{\sqrt{10}}{5}) \cup (- \frac{\sqrt{10}}{5}, \frac{\sqrt{10}}{5}) \cup (\frac{\sqrt{10}}{5}, \infty)$

Set-Builder Notation:

${x | x \neq \frac{\sqrt{10}}{5}, - \frac{\sqrt{10}}{5}}$

Step 4