Precalculus Examples

$\frac{2}{x^{2} - 6 x + y}$

Step 1

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

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

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 $a = 1$ , $b = - 6$ , and $c = y$ into the quadratic formula and solve for $x$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (1 \cdot y)}}{2 \cdot 1}$

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 $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot y}}{2 \cdot 1}$

Step 2.3.1.2

Multiply $- 4$ by $1$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot y}}{2 \cdot 1}$

Step 2.3.1.3

Factor $4$ out of $36 - 4 y$ .

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Step 2.3.1.3.1

Factor $4$ out of $36$ .

$x = \frac{6 \pm \sqrt{4 (9) - 4 y}}{2 \cdot 1}$

Step 2.3.1.3.2

Factor $4$ out of $- 4 y$ .

$x = \frac{6 \pm \sqrt{4 (9) + 4 (- y)}}{2 \cdot 1}$

Step 2.3.1.3.3

Factor $4$ out of $4 (9) + 4 (- y)$ .

$x = \frac{6 \pm \sqrt{4 (9 - y)}}{2 \cdot 1}$

Step 2.3.1.4

Rewrite $4 (9 - y)$ as $2^{2} (3^{2} - y)$ .

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Step 2.3.1.4.1

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

$x = \frac{6 \pm \sqrt{2^{2} (9 - y)}}{2 \cdot 1}$

Step 2.3.1.4.2

Rewrite $9$ as $3^{2}$ .

$x = \frac{6 \pm \sqrt{2^{2} (3^{2} - y)}}{2 \cdot 1}$

Step 2.3.1.5

Pull terms out from under the radical.

$x = \frac{6 \pm 2 \sqrt{3^{2} - y}}{2 \cdot 1}$

Step 2.3.1.6

Raise $3$ to the power of $2$ .

$x = \frac{6 \pm 2 \sqrt{9 - y}}{2 \cdot 1}$

Step 2.3.2

Multiply $2$ by $1$ .

$x = \frac{6 \pm 2 \sqrt{9 - y}}{2}$

Step 2.3.3

Simplify $\frac{6 \pm 2 \sqrt{9 - y}}{2}$ .

$x = 3 \pm \sqrt{9 - y}$

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 $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot y}}{2 \cdot 1}$

Step 2.4.1.2

Multiply $- 4$ by $1$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot y}}{2 \cdot 1}$

Step 2.4.1.3

Factor $4$ out of $36 - 4 y$ .

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Step 2.4.1.3.1

Factor $4$ out of $36$ .

$x = \frac{6 \pm \sqrt{4 (9) - 4 y}}{2 \cdot 1}$

Step 2.4.1.3.2

Factor $4$ out of $- 4 y$ .

$x = \frac{6 \pm \sqrt{4 (9) + 4 (- y)}}{2 \cdot 1}$

Step 2.4.1.3.3

Factor $4$ out of $4 (9) + 4 (- y)$ .

$x = \frac{6 \pm \sqrt{4 (9 - y)}}{2 \cdot 1}$

Step 2.4.1.4

Rewrite $4 (9 - y)$ as $2^{2} (3^{2} - y)$ .

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Step 2.4.1.4.1

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

$x = \frac{6 \pm \sqrt{2^{2} (9 - y)}}{2 \cdot 1}$

Step 2.4.1.4.2

Rewrite $9$ as $3^{2}$ .

$x = \frac{6 \pm \sqrt{2^{2} (3^{2} - y)}}{2 \cdot 1}$

Step 2.4.1.5

Pull terms out from under the radical.

$x = \frac{6 \pm 2 \sqrt{3^{2} - y}}{2 \cdot 1}$

Step 2.4.1.6

Raise $3$ to the power of $2$ .

$x = \frac{6 \pm 2 \sqrt{9 - y}}{2 \cdot 1}$

Step 2.4.2

Multiply $2$ by $1$ .

$x = \frac{6 \pm 2 \sqrt{9 - y}}{2}$

Step 2.4.3

Simplify $\frac{6 \pm 2 \sqrt{9 - y}}{2}$ .

$x = 3 \pm \sqrt{9 - y}$

Step 2.4.4

Change the $\pm$ to $+$ .

$x = 3 + \sqrt{9 - y}$

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 $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot y}}{2 \cdot 1}$

Step 2.5.1.2

Multiply $- 4$ by $1$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot y}}{2 \cdot 1}$

Step 2.5.1.3

Factor $4$ out of $36 - 4 y$ .

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Step 2.5.1.3.1

Factor $4$ out of $36$ .

$x = \frac{6 \pm \sqrt{4 (9) - 4 y}}{2 \cdot 1}$

Step 2.5.1.3.2

Factor $4$ out of $- 4 y$ .

$x = \frac{6 \pm \sqrt{4 (9) + 4 (- y)}}{2 \cdot 1}$

Step 2.5.1.3.3

Factor $4$ out of $4 (9) + 4 (- y)$ .

$x = \frac{6 \pm \sqrt{4 (9 - y)}}{2 \cdot 1}$

Step 2.5.1.4

Rewrite $4 (9 - y)$ as $2^{2} (3^{2} - y)$ .

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Step 2.5.1.4.1

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

$x = \frac{6 \pm \sqrt{2^{2} (9 - y)}}{2 \cdot 1}$

Step 2.5.1.4.2

Rewrite $9$ as $3^{2}$ .

$x = \frac{6 \pm \sqrt{2^{2} (3^{2} - y)}}{2 \cdot 1}$

Step 2.5.1.5

Pull terms out from under the radical.

$x = \frac{6 \pm 2 \sqrt{3^{2} - y}}{2 \cdot 1}$

Step 2.5.1.6

Raise $3$ to the power of $2$ .

$x = \frac{6 \pm 2 \sqrt{9 - y}}{2 \cdot 1}$

Step 2.5.2

Multiply $2$ by $1$ .

$x = \frac{6 \pm 2 \sqrt{9 - y}}{2}$

Step 2.5.3

Simplify $\frac{6 \pm 2 \sqrt{9 - y}}{2}$ .

$x = 3 \pm \sqrt{9 - y}$

Step 2.5.4

Change the $\pm$ to $-$ .

$x = 3 - \sqrt{9 - y}$

Step 2.6

The final answer is the combination of both solutions.

$x = 3 + \sqrt{9 - y}$

$x = 3 - \sqrt{9 - y}$

$x = 3 + \sqrt{9 - y}$

$x = 3 - \sqrt{9 - y}$

Step 3

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$