Precalculus Examples

$f (x) = \frac{1}{x^{2} - 5 x + 25}$

Step 1

Set the denominator in $\frac{1}{x^{2} - 5 x + 25}$ equal to $0$ to find where the expression is undefined.

$x^{2} - 5 x + 25 = 0$

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 $c = 25$ into the quadratic formula and solve for $x$ .

$\frac{5 \pm \sqrt{{(- 5)}^{2} - 4 \cdot (1 \cdot 25)}}{2 \cdot 1}$

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 $2$ .

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

Step 2.3.1.2

Multiply $- 4 \cdot 1 \cdot 25$ .

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Step 2.3.1.2.1

Multiply $- 4$ by $1$ .

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

Step 2.3.1.2.2

Multiply $- 4$ by $25$ .

$x = \frac{5 \pm \sqrt{25 - 100}}{2 \cdot 1}$

Step 2.3.1.3

Subtract $100$ from $25$ .

$x = \frac{5 \pm \sqrt{- 75}}{2 \cdot 1}$

Step 2.3.1.4

Rewrite $- 75$ as $- 1 (75)$ .

$x = \frac{5 \pm \sqrt{- 1 \cdot 75}}{2 \cdot 1}$

Step 2.3.1.5

Rewrite $\sqrt{- 1 (75)}$ as $\sqrt{- 1} \cdot \sqrt{75}$ .

$x = \frac{5 \pm \sqrt{- 1} \cdot \sqrt{75}}{2 \cdot 1}$

Step 2.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{5 \pm i \cdot \sqrt{75}}{2 \cdot 1}$

Step 2.3.1.7

Rewrite $75$ as $5^{2} \cdot 3$ .

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Step 2.3.1.7.1

Factor $25$ out of $75$ .

$x = \frac{5 \pm i \cdot \sqrt{25 (3)}}{2 \cdot 1}$

Step 2.3.1.7.2

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

$x = \frac{5 \pm i \cdot \sqrt{5^{2} \cdot 3}}{2 \cdot 1}$

Step 2.3.1.8

Pull terms out from under the radical.

$x = \frac{5 \pm i \cdot (5 \sqrt{3})}{2 \cdot 1}$

Step 2.3.1.9

Move $5$ to the left of $i$ .

$x = \frac{5 \pm 5 i \sqrt{3}}{2 \cdot 1}$

Step 2.3.2

Multiply $2$ by $1$ .

$x = \frac{5 \pm 5 i \sqrt{3}}{2}$

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 $2$ .

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

Step 2.4.1.2

Multiply $- 4 \cdot 1 \cdot 25$ .

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Step 2.4.1.2.1

Multiply $- 4$ by $1$ .

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

Step 2.4.1.2.2

Multiply $- 4$ by $25$ .

$x = \frac{5 \pm \sqrt{25 - 100}}{2 \cdot 1}$

Step 2.4.1.3

Subtract $100$ from $25$ .

$x = \frac{5 \pm \sqrt{- 75}}{2 \cdot 1}$

Step 2.4.1.4

Rewrite $- 75$ as $- 1 (75)$ .

$x = \frac{5 \pm \sqrt{- 1 \cdot 75}}{2 \cdot 1}$

Step 2.4.1.5

Rewrite $\sqrt{- 1 (75)}$ as $\sqrt{- 1} \cdot \sqrt{75}$ .

$x = \frac{5 \pm \sqrt{- 1} \cdot \sqrt{75}}{2 \cdot 1}$

Step 2.4.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{5 \pm i \cdot \sqrt{75}}{2 \cdot 1}$

Step 2.4.1.7

Rewrite $75$ as $5^{2} \cdot 3$ .

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Step 2.4.1.7.1

Factor $25$ out of $75$ .

$x = \frac{5 \pm i \cdot \sqrt{25 (3)}}{2 \cdot 1}$

Step 2.4.1.7.2

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

$x = \frac{5 \pm i \cdot \sqrt{5^{2} \cdot 3}}{2 \cdot 1}$

Step 2.4.1.8

Pull terms out from under the radical.

$x = \frac{5 \pm i \cdot (5 \sqrt{3})}{2 \cdot 1}$

Step 2.4.1.9

Move $5$ to the left of $i$ .

$x = \frac{5 \pm 5 i \sqrt{3}}{2 \cdot 1}$

Step 2.4.2

Multiply $2$ by $1$ .

$x = \frac{5 \pm 5 i \sqrt{3}}{2}$

Step 2.4.3

Change the $\pm$ to $+$ .

$x = \frac{5 + 5 i \sqrt{3}}{2}$

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 $2$ .

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

Step 2.5.1.2

Multiply $- 4 \cdot 1 \cdot 25$ .

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Step 2.5.1.2.1

Multiply $- 4$ by $1$ .

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

Step 2.5.1.2.2

Multiply $- 4$ by $25$ .

$x = \frac{5 \pm \sqrt{25 - 100}}{2 \cdot 1}$

Step 2.5.1.3

Subtract $100$ from $25$ .

$x = \frac{5 \pm \sqrt{- 75}}{2 \cdot 1}$

Step 2.5.1.4

Rewrite $- 75$ as $- 1 (75)$ .

$x = \frac{5 \pm \sqrt{- 1 \cdot 75}}{2 \cdot 1}$

Step 2.5.1.5

Rewrite $\sqrt{- 1 (75)}$ as $\sqrt{- 1} \cdot \sqrt{75}$ .

$x = \frac{5 \pm \sqrt{- 1} \cdot \sqrt{75}}{2 \cdot 1}$

Step 2.5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{5 \pm i \cdot \sqrt{75}}{2 \cdot 1}$

Step 2.5.1.7

Rewrite $75$ as $5^{2} \cdot 3$ .

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Step 2.5.1.7.1

Factor $25$ out of $75$ .

$x = \frac{5 \pm i \cdot \sqrt{25 (3)}}{2 \cdot 1}$

Step 2.5.1.7.2

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

$x = \frac{5 \pm i \cdot \sqrt{5^{2} \cdot 3}}{2 \cdot 1}$

Step 2.5.1.8

Pull terms out from under the radical.

$x = \frac{5 \pm i \cdot (5 \sqrt{3})}{2 \cdot 1}$

Step 2.5.1.9

Move $5$ to the left of $i$ .

$x = \frac{5 \pm 5 i \sqrt{3}}{2 \cdot 1}$

Step 2.5.2

Multiply $2$ by $1$ .

$x = \frac{5 \pm 5 i \sqrt{3}}{2}$

Step 2.5.3

Change the $\pm$ to $-$ .

$x = \frac{5 - 5 i \sqrt{3}}{2}$

Step 2.6

The final answer is the combination of both solutions.

$x = \frac{5 + 5 i \sqrt{3}}{2}, \frac{5 - 5 i \sqrt{3}}{2}$

Step 3

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 4