Basic Math Examples

$\frac{\frac{14}{x - 9} + \frac{x}{x - 6}}{\frac{3}{x - 10} - \frac{2}{x - 9}}$

Step 1

Multiply the numerator and denominator of the fraction by $(x - 9) (x - 6) (x - 10)$ .

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Step 1.1

Multiply $\frac{\frac{14}{x - 9} + \frac{x}{x - 6}}{\frac{3}{x - 10} - \frac{2}{x - 9}}$ by $\frac{(x - 9) (x - 6) (x - 10)}{(x - 9) (x - 6) (x - 10)}$ .

$\frac{(x - 9) (x - 6) (x - 10)}{(x - 9) (x - 6) (x - 10)} \cdot \frac{\frac{14}{x - 9} + \frac{x}{x - 6}}{\frac{3}{x - 10} - \frac{2}{x - 9}}$

Step 1.2

Combine.

$\frac{(x - 9) (x - 6) (x - 10) (\frac{14}{x - 9} + \frac{x}{x - 6})}{(x - 9) (x - 6) (x - 10) (\frac{3}{x - 10} - \frac{2}{x - 9})}$

Step 2

Apply the distributive property.

$\frac{(x - 9) (x - 6) (x - 10) \frac{14}{x - 9} + (x - 9) (x - 6) (x - 10) \frac{x}{x - 6}}{(x - 9) (x - 6) (x - 10) \frac{3}{x - 10} + (x - 9) (x - 6) (x - 10) (- \frac{2}{x - 9})}$

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $x - 9$ .

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Step 3.1.1

Factor $x - 9$ out of $(x - 9) (x - 6) (x - 10)$ .

$\frac{(x - 9) ((x - 6) (x - 10)) \frac{14}{x - 9} + (x - 9) (x - 6) (x - 10) \frac{x}{x - 6}}{(x - 9) (x - 6) (x - 10) \frac{3}{x - 10} + (x - 9) (x - 6) (x - 10) (- \frac{2}{x - 9})}$

Step 3.1.2

Cancel the common factor.

$\frac{(x - 9) ((x - 6) (x - 10)) \frac{14}{x - 9} + (x - 9) (x - 6) (x - 10) \frac{x}{x - 6}}{(x - 9) (x - 6) (x - 10) \frac{3}{x - 10} + (x - 9) (x - 6) (x - 10) (- \frac{2}{x - 9})}$

Step 3.1.3

Rewrite the expression.

$\frac{(x - 6) (x - 10) \cdot 14 + (x - 9) (x - 6) (x - 10) \frac{x}{x - 6}}{(x - 9) (x - 6) (x - 10) \frac{3}{x - 10} + (x - 9) (x - 6) (x - 10) (- \frac{2}{x - 9})}$

Step 3.2

Cancel the common factor of $x - 6$ .

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Step 3.2.1

Factor $x - 6$ out of $(x - 9) (x - 6) (x - 10)$ .

$\frac{(x - 6) (x - 10) \cdot 14 + (x - 6) ((x - 9) (x - 10)) \frac{x}{x - 6}}{(x - 9) (x - 6) (x - 10) \frac{3}{x - 10} + (x - 9) (x - 6) (x - 10) (- \frac{2}{x - 9})}$

Step 3.2.2

Cancel the common factor.

$\frac{(x - 6) (x - 10) \cdot 14 + (x - 6) ((x - 9) (x - 10)) \frac{x}{x - 6}}{(x - 9) (x - 6) (x - 10) \frac{3}{x - 10} + (x - 9) (x - 6) (x - 10) (- \frac{2}{x - 9})}$

Step 3.2.3

Rewrite the expression.

$\frac{(x - 6) (x - 10) \cdot 14 + (x - 9) (x - 10) x}{(x - 9) (x - 6) (x - 10) \frac{3}{x - 10} + (x - 9) (x - 6) (x - 10) (- \frac{2}{x - 9})}$

Step 3.3

Cancel the common factor of $x - 10$ .

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Step 3.3.1

Factor $x - 10$ out of $(x - 9) (x - 6) (x - 10)$ .

$\frac{(x - 6) (x - 10) \cdot 14 + (x - 9) (x - 10) x}{(x - 10) ((x - 9) (x - 6)) \frac{3}{x - 10} + (x - 9) (x - 6) (x - 10) (- \frac{2}{x - 9})}$

Step 3.3.2

Cancel the common factor.

$\frac{(x - 6) (x - 10) \cdot 14 + (x - 9) (x - 10) x}{(x - 10) ((x - 9) (x - 6)) \frac{3}{x - 10} + (x - 9) (x - 6) (x - 10) (- \frac{2}{x - 9})}$

Step 3.3.3

Rewrite the expression.

$\frac{(x - 6) (x - 10) \cdot 14 + (x - 9) (x - 10) x}{(x - 9) (x - 6) \cdot 3 + (x - 9) (x - 6) (x - 10) (- \frac{2}{x - 9})}$

Step 3.4

Cancel the common factor of $x - 9$ .

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Step 3.4.1

Move the leading negative in $- \frac{2}{x - 9}$ into the numerator.

$\frac{(x - 6) (x - 10) \cdot 14 + (x - 9) (x - 10) x}{(x - 9) (x - 6) \cdot 3 + (x - 9) (x - 6) (x - 10) \frac{- 2}{x - 9}}$

Step 3.4.2

Factor $x - 9$ out of $(x - 9) (x - 6) (x - 10)$ .

$\frac{(x - 6) (x - 10) \cdot 14 + (x - 9) (x - 10) x}{(x - 9) (x - 6) \cdot 3 + (x - 9) ((x - 6) (x - 10)) \frac{- 2}{x - 9}}$

Step 3.4.3

Cancel the common factor.

$\frac{(x - 6) (x - 10) \cdot 14 + (x - 9) (x - 10) x}{(x - 9) (x - 6) \cdot 3 + (x - 9) ((x - 6) (x - 10)) \frac{- 2}{x - 9}}$

Step 3.4.4

Rewrite the expression.

$\frac{(x - 6) (x - 10) \cdot 14 + (x - 9) (x - 10) x}{(x - 9) (x - 6) \cdot 3 + (x - 6) (x - 10) \cdot - 2}$

Step 4

Simplify the numerator.

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Step 4.1

Factor $x - 10$ out of $(x - 6) (x - 10) \cdot 14 + (x - 9) (x - 10) x$ .

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Step 4.1.1

Factor $x - 10$ out of $(x - 6) (x - 10) \cdot 14$ .

$\frac{(x - 10) ((x - 6) \cdot 14) + (x - 9) (x - 10) x}{(x - 9) (x - 6) \cdot 3 + (x - 6) (x - 10) \cdot - 2}$

Step 4.1.2

Factor $x - 10$ out of $(x - 9) (x - 10) x$ .

$\frac{(x - 10) ((x - 6) \cdot 14) + (x - 10) ((x - 9) x)}{(x - 9) (x - 6) \cdot 3 + (x - 6) (x - 10) \cdot - 2}$

Step 4.1.3

Factor $x - 10$ out of $(x - 10) ((x - 6) \cdot 14) + (x - 10) ((x - 9) x)$ .

$\frac{(x - 10) ((x - 6) \cdot 14 + (x - 9) x)}{(x - 9) (x - 6) \cdot 3 + (x - 6) (x - 10) \cdot - 2}$

Step 4.2

Apply the distributive property.

$\frac{(x - 10) (x \cdot 14 - 6 \cdot 14 + (x - 9) x)}{(x - 9) (x - 6) \cdot 3 + (x - 6) (x - 10) \cdot - 2}$

Step 4.3

Move $14$ to the left of $x$ .

$\frac{(x - 10) (14 \cdot x - 6 \cdot 14 + (x - 9) x)}{(x - 9) (x - 6) \cdot 3 + (x - 6) (x - 10) \cdot - 2}$

Step 4.4

Multiply $- 6$ by $14$ .

$\frac{(x - 10) (14 \cdot x - 84 + (x - 9) x)}{(x - 9) (x - 6) \cdot 3 + (x - 6) (x - 10) \cdot - 2}$

Step 4.5

Apply the distributive property.

$\frac{(x - 10) (14 x - 84 + x \cdot x - 9 x)}{(x - 9) (x - 6) \cdot 3 + (x - 6) (x - 10) \cdot - 2}$

Step 4.6

Multiply $x$ by $x$ .

$\frac{(x - 10) (14 x - 84 + x^{2} - 9 x)}{(x - 9) (x - 6) \cdot 3 + (x - 6) (x - 10) \cdot - 2}$

Step 4.7

Subtract $9 x$ from $14 x$ .

$\frac{(x - 10) (5 x - 84 + x^{2})}{(x - 9) (x - 6) \cdot 3 + (x - 6) (x - 10) \cdot - 2}$

Step 4.8

Reorder terms.

$\frac{(x - 10) (x^{2} + 5 x - 84)}{(x - 9) (x - 6) \cdot 3 + (x - 6) (x - 10) \cdot - 2}$

Step 4.9

Factor $x^{2} + 5 x - 84$ using the AC method.

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Step 4.9.1

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 $- 84$ and whose sum is $5$ .

$- 7, 12$

Step 4.9.2

Write the factored form using these integers.

$\frac{(x - 10) ((x - 7) (x + 12))}{(x - 9) (x - 6) \cdot 3 + (x - 6) (x - 10) \cdot - 2}$

$\frac{(x - 10) (x - 7) (x + 12)}{(x - 9) (x - 6) \cdot 3 + (x - 6) (x - 10) \cdot - 2}$

Step 5

Simplify the denominator.

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Step 5.1

Factor $x - 6$ out of $(x - 9) (x - 6) \cdot 3 + (x - 6) (x - 10) \cdot - 2$ .

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Step 5.1.1

Factor $x - 6$ out of $(x - 9) (x - 6) \cdot 3$ .

$\frac{(x - 10) (x - 7) (x + 12)}{(x - 6) ((x - 9) \cdot 3) + (x - 6) (x - 10) \cdot - 2}$

Step 5.1.2

Factor $x - 6$ out of $(x - 6) (x - 10) \cdot - 2$ .

$\frac{(x - 10) (x - 7) (x + 12)}{(x - 6) ((x - 9) \cdot 3) + (x - 6) ((x - 10) \cdot - 2)}$

Step 5.1.3

Factor $x - 6$ out of $(x - 6) ((x - 9) \cdot 3) + (x - 6) ((x - 10) \cdot - 2)$ .

$\frac{(x - 10) (x - 7) (x + 12)}{(x - 6) ((x - 9) \cdot 3 + (x - 10) \cdot - 2)}$

Step 5.2

Apply the distributive property.

$\frac{(x - 10) (x - 7) (x + 12)}{(x - 6) (x \cdot 3 - 9 \cdot 3 + (x - 10) \cdot - 2)}$

Step 5.3

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

$\frac{(x - 10) (x - 7) (x + 12)}{(x - 6) (3 \cdot x - 9 \cdot 3 + (x - 10) \cdot - 2)}$

Step 5.4

Multiply $- 9$ by $3$ .

$\frac{(x - 10) (x - 7) (x + 12)}{(x - 6) (3 \cdot x - 27 + (x - 10) \cdot - 2)}$

Step 5.5

Apply the distributive property.

$\frac{(x - 10) (x - 7) (x + 12)}{(x - 6) (3 x - 27 + x \cdot - 2 - 10 \cdot - 2)}$

Step 5.6

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

$\frac{(x - 10) (x - 7) (x + 12)}{(x - 6) (3 x - 27 - 2 \cdot x - 10 \cdot - 2)}$

Step 5.7

Multiply $- 10$ by $- 2$ .

$\frac{(x - 10) (x - 7) (x + 12)}{(x - 6) (3 x - 27 - 2 \cdot x + 20)}$

Step 5.8

Subtract $2 x$ from $3 x$ .

$\frac{(x - 10) (x - 7) (x + 12)}{(x - 6) (x - 27 + 20)}$

Step 5.9

Add $- 27$ and $20$ .

$\frac{(x - 10) (x - 7) (x + 12)}{(x - 6) (x - 7)}$

Step 6

Cancel the common factor of $x - 7$ .

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Step 6.1

Cancel the common factor.

$\frac{(x - 10) (x - 7) (x + 12)}{(x - 6) (x - 7)}$

Step 6.2

Rewrite the expression.

$\frac{(x - 10) (x + 12)}{x - 6}$