Algebra Examples

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

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.

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

Step 2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Expand $(x + 1) (x - 5)$ using the FOIL Method.

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Step 5.1.1

Apply the distributive property.

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

Step 5.1.2

Apply the distributive property.

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

Step 5.1.3

Apply the distributive property.

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

Step 5.2

Simplify and combine like terms.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Multiply $x$ by $x$ .

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

Step 5.2.1.2

Move $- 5$ to the left of $x$ .

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

Step 5.2.1.3

Multiply $x$ by $1$ .

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

Step 5.2.1.4

Multiply $- 5$ by $1$ .

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

Step 5.2.2

Add $- 5 x$ and $x$ .

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

Step 5.3

Subtract $4 x$ from $2 x$ .

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

Step 5.4

Subtract $5$ from $5$ .

$\frac{- 2 x + x^{2} + 0}{(x + 2) (x - 5)}$

Step 5.5

Add $- 2 x + x^{2}$ and $0$ .

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

Step 5.6

Factor $x$ out of $- 2 x + x^{2}$ .

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Step 5.6.1

Factor $x$ out of $- 2 x$ .

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

Step 5.6.2

Factor $x$ out of $x^{2}$ .

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

Step 5.6.3

Factor $x$ out of $x \cdot - 2 + x \cdot x$ .

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

Step 6

Simplify with factoring out.

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Step 6.1

Rewrite $- 2$ as $- 1 (2)$ .

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

Step 6.2

Factor $- 1$ out of $x$ .

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

Step 6.3

Factor $- 1$ out of $- 1 (2) - 1 (- x)$ .

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

Step 6.4

Move the negative in front of the fraction.

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