Simplify (x+1)/(x^2-4)+(2x-3)/(x^2+x-2)

$\frac{x + 1}{x^{2} - 4} + \frac{2 x - 3}{x^{2} + x - 2}$

Step 1

Simplify each term.

Simplify the denominator.

Rewrite $4$ as $2^{2}$ .

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

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 $b = 2$ .

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

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

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 $- 2$ and whose sum is $1$ .

$- 1, 2$

Write the factored form using these integers.

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

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

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

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

Reorder the factors of $(x + 2) (x - 2) (x - 1)$ .

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

Reorder the factors of $(x - 1) (x + 2) (x - 2)$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Move $- 1$ to the left of $x$ .

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

Rewrite $- 1 x$ as $- x$ .

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

Multiply $x$ by $1$ .

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

Multiply $- 1$ by $1$ .

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

Add $- x$ and $x$ .

$\frac{x^{2} + 0 - 1 + (2 x - 3) (x - 2)}{(x + 2) (x - 1) (x - 2)}$

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

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

Expand $(2 x - 3) (x - 2)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 2$ by $2$ .

$\frac{x^{2} - 1 + 2 x^{2} - 4 x - 3 x - 3 \cdot - 2}{(x + 2) (x - 1) (x - 2)}$

Multiply $- 3$ by $- 2$ .

$\frac{x^{2} - 1 + 2 x^{2} - 4 x - 3 x + 6}{(x + 2) (x - 1) (x - 2)}$

Subtract $3 x$ from $- 4 x$ .

$\frac{x^{2} - 1 + 2 x^{2} - 7 x + 6}{(x + 2) (x - 1) (x - 2)}$

Add $x^{2}$ and $2 x^{2}$ .

$\frac{3 x^{2} - 1 - 7 x + 6}{(x + 2) (x - 1) (x - 2)}$

Add $- 1$ and $6$ .

$\frac{3 x^{2} - 7 x + 5}{(x + 2) (x - 1) (x - 2)}$

Simplify (x+1)/(x^2-4)+(2x-3)/(x^2+x-2)

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