Calculus Examples

$\frac{x}{x - 3} + \frac{x - 21}{x^{2} - 9}$

Step 1

Simplify the denominator.

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Step 1.1

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

$\frac{x}{x - 3} + \frac{x - 21}{x^{2} - 3^{2}}$

Step 1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 3$ .

$\frac{x}{x - 3} + \frac{x - 21}{(x + 3) (x - 3)}$

Step 2

To write $\frac{x}{x - 3}$ as a fraction with a common denominator, multiply by $\frac{x + 3}{x + 3}$ .

$\frac{x}{x - 3} \cdot \frac{x + 3}{x + 3} + \frac{x - 21}{(x + 3) (x - 3)}$

Step 3

Write each expression with a common denominator of $(x - 3) (x + 3)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

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

$\frac{x (x + 3)}{(x - 3) (x + 3)} + \frac{x - 21}{(x + 3) (x - 3)}$

Step 3.2

Reorder the factors of $(x - 3) (x + 3)$ .

$\frac{x (x + 3)}{(x + 3) (x - 3)} + \frac{x - 21}{(x + 3) (x - 3)}$

Step 4

Combine the numerators over the common denominator.

$\frac{x (x + 3) + x - 21}{(x + 3) (x - 3)}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

$\frac{x \cdot x + x \cdot 3 + x - 21}{(x + 3) (x - 3)}$

Step 5.2

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot 3 + x - 21}{(x + 3) (x - 3)}$

Step 5.3

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

$\frac{x^{2} + 3 \cdot x + x - 21}{(x + 3) (x - 3)}$

Step 5.4

Add $3 x$ and $x$ .

$\frac{x^{2} + 4 x - 21}{(x + 3) (x - 3)}$

Step 5.5

Factor $x^{2} + 4 x - 21$ using the AC method.

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Step 5.5.1

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 $- 21$ and whose sum is $4$ .

$- 3, 7$

Step 5.5.2

Write the factored form using these integers.

$\frac{(x - 3) (x + 7)}{(x + 3) (x - 3)}$

Step 6

Cancel the common factor of $x - 3$ .

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Step 6.1

Cancel the common factor.

$\frac{(x - 3) (x + 7)}{(x + 3) (x - 3)}$

Step 6.2

Rewrite the expression.

$\frac{x + 7}{x + 3}$