Precalculus Examples

$\frac{a^{2}}{a x - x^{2}} + \frac{x}{x - a}$

Step 1

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

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

Step 2

Solve for $a$ .

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Step 2.1

Add $x^{2}$ to both sides of the equation.

$a x = x^{2}$

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $x$ .

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Step 2.2.2.1.1

Cancel the common factor.

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

Step 2.2.2.1.2

Divide $a$ by $1$ .

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

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Cancel the common factor of $x^{2}$ and $x$ .

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Step 2.2.3.1.1

Factor $x$ out of $x^{2}$ .

$a = \frac{x \cdot x}{x}$

Step 2.2.3.1.2

Cancel the common factors.

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Step 2.2.3.1.2.1

Raise $x$ to the power of $1$ .

$a = \frac{x \cdot x}{x^{1}}$

Step 2.2.3.1.2.2

Factor $x$ out of $x^{1}$ .

$a = \frac{x \cdot x}{x \cdot 1}$

Step 2.2.3.1.2.3

Cancel the common factor.

$a = \frac{x \cdot x}{x \cdot 1}$

Step 2.2.3.1.2.4

Rewrite the expression.

$a = \frac{x}{1}$

Step 2.2.3.1.2.5

Divide $x$ by $1$ .

$a = x$

Step 3

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

$x - a = 0$

Step 4

Solve for $a$ .

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Step 4.1

Subtract $x$ from both sides of the equation.

$- a = - x$

Step 4.2

Divide each term in $- a = - x$ by $- 1$ and simplify.

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Step 4.2.1

Divide each term in $- a = - x$ by $- 1$ .

$\frac{- a}{- 1} = \frac{- x}{- 1}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Dividing two negative values results in a positive value.

$\frac{a}{1} = \frac{- x}{- 1}$

Step 4.2.2.2

Divide $a$ by $1$ .

$a = \frac{- x}{- 1}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Dividing two negative values results in a positive value.

$a = \frac{x}{1}$

Step 4.2.3.2

Divide $x$ by $1$ .

$a = x$

Step 5

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${a | a \in ℝ}$