Linear Algebra Examples

$\frac{- 3}{x^{2} - 5 x - 3}$

Step 1

Set the denominator in $\frac{- 3}{x^{2} - 5 x - 3}$ equal to $0$ to find where the expression is undefined.

$x^{2} - 5 x - 3 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2.2

Substitute the values $a = 1$ , $b = - 5$ , and $c = - 3$ into the quadratic formula and solve for $x$ .

$\frac{5 \pm \sqrt{{(- 5)}^{2} - 4 \cdot (1 \cdot - 3)}}{2 \cdot 1}$

Step 2.3

Simplify.

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Step 2.3.1

Simplify the numerator.

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Step 2.3.1.1

Raise $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 1 \cdot - 3}}{2 \cdot 1}$

Step 2.3.1.2

Multiply $- 4 \cdot 1 \cdot - 3$ .

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Step 2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot - 3}}{2 \cdot 1}$

Step 2.3.1.2.2

Multiply $- 4$ by $- 3$ .

$x = \frac{5 \pm \sqrt{25 + 12}}{2 \cdot 1}$

Step 2.3.1.3

Add $25$ and $12$ .

$x = \frac{5 \pm \sqrt{37}}{2 \cdot 1}$

Step 2.3.2

Multiply $2$ by $1$ .

$x = \frac{5 \pm \sqrt{37}}{2}$

Step 2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 2.4.1

Simplify the numerator.

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Step 2.4.1.1

Raise $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 1 \cdot - 3}}{2 \cdot 1}$

Step 2.4.1.2

Multiply $- 4 \cdot 1 \cdot - 3$ .

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Step 2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot - 3}}{2 \cdot 1}$

Step 2.4.1.2.2

Multiply $- 4$ by $- 3$ .

$x = \frac{5 \pm \sqrt{25 + 12}}{2 \cdot 1}$

Step 2.4.1.3

Add $25$ and $12$ .

$x = \frac{5 \pm \sqrt{37}}{2 \cdot 1}$

Step 2.4.2

Multiply $2$ by $1$ .

$x = \frac{5 \pm \sqrt{37}}{2}$

Step 2.4.3

Change the $\pm$ to $+$ .

$x = \frac{5 + \sqrt{37}}{2}$

Step 2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

Raise $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 1 \cdot - 3}}{2 \cdot 1}$

Step 2.5.1.2

Multiply $- 4 \cdot 1 \cdot - 3$ .

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Step 2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot - 3}}{2 \cdot 1}$

Step 2.5.1.2.2

Multiply $- 4$ by $- 3$ .

$x = \frac{5 \pm \sqrt{25 + 12}}{2 \cdot 1}$

Step 2.5.1.3

Add $25$ and $12$ .

$x = \frac{5 \pm \sqrt{37}}{2 \cdot 1}$

Step 2.5.2

Multiply $2$ by $1$ .

$x = \frac{5 \pm \sqrt{37}}{2}$

Step 2.5.3

Change the $\pm$ to $-$ .

$x = \frac{5 - \sqrt{37}}{2}$

Step 2.6

The final answer is the combination of both solutions.

$x = \frac{5 + \sqrt{37}}{2}, \frac{5 - \sqrt{37}}{2}$

Step 3

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

Interval Notation:

$(- \infty, \frac{5 - \sqrt{37}}{2}) \cup (\frac{5 - \sqrt{37}}{2}, \frac{5 + \sqrt{37}}{2}) \cup (\frac{5 + \sqrt{37}}{2}, \infty)$

Set-Builder Notation:

${x | x \neq \frac{5 - \sqrt{37}}{2}, \frac{5 + \sqrt{37}}{2}}$

Step 4