Finite Math Examples

$\frac{x + 1}{x^{2} - 3 x - 10} + \frac{x}{x - 5}$

Step 1

Factor $x^{2} - 3 x - 10$ using the AC method.

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Step 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 $- 3$ .

$- 5, 2$

Step 1.2

Write the factored form using these integers.

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

Step 2

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

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

Step 3

Simplify terms.

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Step 3.1

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

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Multiply $x$ by $x$ .

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

Step 4.3

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

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

Step 4.4

Add $x$ and $2 x$ .

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

Step 4.5

Reorder terms.

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