Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

Factor $x^{2} - 4 x - 5$ 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 $- 5$ and whose sum is $- 4$ .

$- 5, 1$

Step 1.1.2

Write the factored form using these integers.

$\frac{2 x}{(x - 5) (x + 1)} - \frac{3}{x^{2} - 2 x - 3}$

Step 1.2

Factor $x^{2} - 2 x - 3$ using the AC method.

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Step 1.2.1

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 $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Step 1.2.2

Write the factored form using these integers.

$\frac{2 x}{(x - 5) (x + 1)} - \frac{3}{(x - 3) (x + 1)}$

Step 2

To write $\frac{2 x}{(x - 5) (x + 1)}$ as a fraction with a common denominator, multiply by $\frac{x - 3}{x - 3}$ .

$\frac{2 x}{(x - 5) (x + 1)} \cdot \frac{x - 3}{x - 3} - \frac{3}{(x - 3) (x + 1)}$

Step 3

To write $- \frac{3}{(x - 3) (x + 1)}$ as a fraction with a common denominator, multiply by $\frac{x - 5}{x - 5}$ .

$\frac{2 x}{(x - 5) (x + 1)} \cdot \frac{x - 3}{x - 3} - \frac{3}{(x - 3) (x + 1)} \cdot \frac{x - 5}{x - 5}$

Step 4

Write each expression with a common denominator of $(x - 5) (x + 1) (x - 3)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{2 x}{(x - 5) (x + 1)}$ by $\frac{x - 3}{x - 3}$ .

$\frac{2 x (x - 3)}{(x - 5) (x + 1) (x - 3)} - \frac{3}{(x - 3) (x + 1)} \cdot \frac{x - 5}{x - 5}$

Step 4.2

Multiply $\frac{3}{(x - 3) (x + 1)}$ by $\frac{x - 5}{x - 5}$ .

$\frac{2 x (x - 3)}{(x - 5) (x + 1) (x - 3)} - \frac{3 (x - 5)}{(x - 3) (x + 1) (x - 5)}$

Step 4.3

Reorder the factors of $(x - 5) (x + 1) (x - 3)$ .

$\frac{2 x (x - 3)}{(x + 1) (x - 3) (x - 5)} - \frac{3 (x - 5)}{(x - 3) (x + 1) (x - 5)}$

Step 4.4

Reorder the factors of $(x - 3) (x + 1) (x - 5)$ .

$\frac{2 x (x - 3)}{(x + 1) (x - 3) (x - 5)} - \frac{3 (x - 5)}{(x + 1) (x - 3) (x - 5)}$

Step 5

Combine the numerators over the common denominator.

$\frac{2 x (x - 3) - 3 (x - 5)}{(x + 1) (x - 3) (x - 5)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{2 x \cdot x + 2 x \cdot - 3 - 3 (x - 5)}{(x + 1) (x - 3) (x - 5)}$

Step 6.2

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

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Step 6.2.1

Move $x$ .

$\frac{2 (x \cdot x) + 2 x \cdot - 3 - 3 (x - 5)}{(x + 1) (x - 3) (x - 5)}$

Step 6.2.2

Multiply $x$ by $x$ .

$\frac{2 x^{2} + 2 x \cdot - 3 - 3 (x - 5)}{(x + 1) (x - 3) (x - 5)}$

Step 6.3

Multiply $- 3$ by $2$ .

$\frac{2 x^{2} - 6 x - 3 (x - 5)}{(x + 1) (x - 3) (x - 5)}$

Step 6.4

Apply the distributive property.

$\frac{2 x^{2} - 6 x - 3 x - 3 \cdot - 5}{(x + 1) (x - 3) (x - 5)}$

Step 6.5

Multiply $- 3$ by $- 5$ .

$\frac{2 x^{2} - 6 x - 3 x + 15}{(x + 1) (x - 3) (x - 5)}$

Step 6.6

Subtract $3 x$ from $- 6 x$ .

$\frac{2 x^{2} - 9 x + 15}{(x + 1) (x - 3) (x - 5)}$