Calculus Examples

$\frac{5 x}{x^{2} - 7 x + 10} - \frac{4}{x^{2} - 25}$

Step 1

Simplify each term.

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Step 1.1

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

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Step 1.1.1

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 1.1.2

Write the factored form using these integers.

$\frac{5 x}{(x - 5) (x - 2)} - \frac{4}{x^{2} - 25}$

Step 1.2

Simplify the denominator.

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Step 1.2.1

Rewrite $25$ as $5^{2}$ .

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

Step 1.2.2

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 $b = 5$ .

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

Step 2

To write $\frac{5 x}{(x - 5) (x - 2)}$ as a fraction with a common denominator, multiply by $\frac{x + 5}{x + 5}$ .

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

Step 3

To write $- \frac{4}{(x + 5) (x - 5)}$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

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

Step 4

Write each expression with a common denominator of $(x - 5) (x - 2) (x + 5)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{5 x}{(x - 5) (x - 2)}$ by $\frac{x + 5}{x + 5}$ .

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

Step 4.2

Multiply $\frac{4}{(x + 5) (x - 5)}$ by $\frac{x - 2}{x - 2}$ .

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

Step 4.3

Reorder the factors of $(x - 5) (x - 2) (x + 5)$ .

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

Step 4.4

Reorder the factors of $(x + 5) (x - 5) (x - 2)$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Multiply $x$ by $x$ by adding the exponents.

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Step 6.2.1

Move $x$ .

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

Step 6.2.2

Multiply $x$ by $x$ .

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

Step 6.3

Multiply $5$ by $5$ .

$\frac{5 x^{2} + 25 x - 4 (x - 2)}{(x + 5) (x - 2) (x - 5)}$

Step 6.4

Apply the distributive property.

$\frac{5 x^{2} + 25 x - 4 x - 4 \cdot - 2}{(x + 5) (x - 2) (x - 5)}$

Step 6.5

Multiply $- 4$ by $- 2$ .

$\frac{5 x^{2} + 25 x - 4 x + 8}{(x + 5) (x - 2) (x - 5)}$

Step 6.6

Subtract $4 x$ from $25 x$ .

$\frac{5 x^{2} + 21 x + 8}{(x + 5) (x - 2) (x - 5)}$