Finite Math Examples

$\frac{x + 2 + \frac{3}{x - 4}}{x - 3 - \frac{2}{x - 4}}$

Step 1

Multiply the numerator and denominator of the fraction by $x - 4$ .

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Step 1.1

Multiply $\frac{x + 2 + \frac{3}{x - 4}}{x - 3 - \frac{2}{x - 4}}$ by $\frac{x - 4}{x - 4}$ .

$\frac{x - 4}{x - 4} \cdot \frac{x + 2 + \frac{3}{x - 4}}{x - 3 - \frac{2}{x - 4}}$

Step 1.2

Combine.

$\frac{(x - 4) (x + 2 + \frac{3}{x - 4})}{(x - 4) (x - 3 - \frac{2}{x - 4})}$

Step 2

Apply the distributive property.

$\frac{(x - 4) x + (x - 4) \cdot 2 + (x - 4) \frac{3}{x - 4}}{(x - 4) x + (x - 4) \cdot - 3 + (x - 4) (- \frac{2}{x - 4})}$

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $x - 4$ .

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Step 3.1.1

Cancel the common factor.

$\frac{(x - 4) x + (x - 4) \cdot 2 + (x - 4) \frac{3}{x - 4}}{(x - 4) x + (x - 4) \cdot - 3 + (x - 4) (- \frac{2}{x - 4})}$

Step 3.1.2

Rewrite the expression.

$\frac{(x - 4) x + (x - 4) \cdot 2 + 3}{(x - 4) x + (x - 4) \cdot - 3 + (x - 4) (- \frac{2}{x - 4})}$

Step 3.2

Cancel the common factor of $x - 4$ .

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Step 3.2.1

Move the leading negative in $- \frac{2}{x - 4}$ into the numerator.

$\frac{(x - 4) x + (x - 4) \cdot 2 + 3}{(x - 4) x + (x - 4) \cdot - 3 + (x - 4) \frac{- 2}{x - 4}}$

Step 3.2.2

Cancel the common factor.

$\frac{(x - 4) x + (x - 4) \cdot 2 + 3}{(x - 4) x + (x - 4) \cdot - 3 + (x - 4) \frac{- 2}{x - 4}}$

Step 3.2.3

Rewrite the expression.

$\frac{(x - 4) x + (x - 4) \cdot 2 + 3}{(x - 4) x + (x - 4) \cdot - 3 - 2}$

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

$\frac{x \cdot x - 4 x + (x - 4) \cdot 2 + 3}{(x - 4) x + (x - 4) \cdot - 3 - 2}$

Step 4.2

Multiply $x$ by $x$ .

$\frac{x^{2} - 4 x + (x - 4) \cdot 2 + 3}{(x - 4) x + (x - 4) \cdot - 3 - 2}$

Step 4.3

Apply the distributive property.

$\frac{x^{2} - 4 x + x \cdot 2 - 4 \cdot 2 + 3}{(x - 4) x + (x - 4) \cdot - 3 - 2}$

Step 4.4

Move $2$ to the left of $x$ .

$\frac{x^{2} - 4 x + 2 \cdot x - 4 \cdot 2 + 3}{(x - 4) x + (x - 4) \cdot - 3 - 2}$

Step 4.5

Multiply $- 4$ by $2$ .

$\frac{x^{2} - 4 x + 2 \cdot x - 8 + 3}{(x - 4) x + (x - 4) \cdot - 3 - 2}$

Step 4.6

Add $- 4 x$ and $2 x$ .

$\frac{x^{2} - 2 x - 8 + 3}{(x - 4) x + (x - 4) \cdot - 3 - 2}$

Step 4.7

Add $- 8$ and $3$ .

$\frac{x^{2} - 2 x - 5}{(x - 4) x + (x - 4) \cdot - 3 - 2}$

Step 5

Simplify the denominator.

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Step 5.1

Apply the distributive property.

$\frac{x^{2} - 2 x - 5}{x \cdot x - 4 x + (x - 4) \cdot - 3 - 2}$

Step 5.2

Multiply $x$ by $x$ .

$\frac{x^{2} - 2 x - 5}{x^{2} - 4 x + (x - 4) \cdot - 3 - 2}$

Step 5.3

Apply the distributive property.

$\frac{x^{2} - 2 x - 5}{x^{2} - 4 x + x \cdot - 3 - 4 \cdot - 3 - 2}$

Step 5.4

Move $- 3$ to the left of $x$ .

$\frac{x^{2} - 2 x - 5}{x^{2} - 4 x - 3 \cdot x - 4 \cdot - 3 - 2}$

Step 5.5

Multiply $- 4$ by $- 3$ .

$\frac{x^{2} - 2 x - 5}{x^{2} - 4 x - 3 \cdot x + 12 - 2}$

Step 5.6

Subtract $3 x$ from $- 4 x$ .

$\frac{x^{2} - 2 x - 5}{x^{2} - 7 x + 12 - 2}$

Step 5.7

Subtract $2$ from $12$ .

$\frac{x^{2} - 2 x - 5}{x^{2} - 7 x + 10}$

Step 5.8

Factor $x^{2} - 7 x + 10$ using the AC method.

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Step 5.8.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $10$ and whose sum is $- 7$ .

$- 5, - 2$

Step 5.8.2

Write the factored form using these integers.

$\frac{x^{2} - 2 x - 5}{(x - 5) (x - 2)}$