Simplify (a+6)/(4-a^2)-(a+3)/(a+2)+(a-3)/(2-a)

$\frac{a + 6}{4 - a^{2}} - \frac{a + 3}{a + 2} + \frac{a - 3}{2 - a}$

Step 1

Simplify the denominator.

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

$\frac{a + 6}{2^{2} - a^{2}} - \frac{a + 3}{a + 2} + \frac{a - 3}{2 - a}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2$ and $b = a$ .

$\frac{a + 6}{(2 + a) (2 - a)} - \frac{a + 3}{a + 2} + \frac{a - 3}{2 - a}$

Step 2

Reorder terms.

$\frac{a + 6}{(a + 2) (2 - a)} - \frac{a + 3}{a + 2} + \frac{a - 3}{2 - a}$

Step 3

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

$\frac{a + 6}{(a + 2) (2 - a)} - \frac{a + 3}{a + 2} \cdot \frac{2 - a}{2 - a} + \frac{a - 3}{2 - a}$

Step 4

Simplify terms.

Multiply $\frac{a + 3}{a + 2}$ by $\frac{2 - a}{2 - a}$ .

$\frac{a + 6}{(a + 2) (2 - a)} - \frac{(a + 3) (2 - a)}{(a + 2) (2 - a)} + \frac{a - 3}{2 - a}$

Combine the numerators over the common denominator.

$\frac{a + 6 - (a + 3) (2 - a)}{(a + 2) (2 - a)} + \frac{a - 3}{2 - a}$

Step 5

Simplify each term.

Simplify the numerator.

Apply the distributive property.

$\frac{a + 6 + (- a - 1 \cdot 3) (2 - a)}{(a + 2) (2 - a)} + \frac{a - 3}{2 - a}$

Multiply $- 1$ by $3$ .

$\frac{a + 6 + (- a - 3) (2 - a)}{(a + 2) (2 - a)} + \frac{a - 3}{2 - a}$

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

Apply the distributive property.

$\frac{a + 6 - a (2 - a) - 3 (2 - a)}{(a + 2) (2 - a)} + \frac{a - 3}{2 - a}$

Apply the distributive property.

$\frac{a + 6 - a \cdot 2 - a (- a) - 3 (2 - a)}{(a + 2) (2 - a)} + \frac{a - 3}{2 - a}$

Apply the distributive property.

$\frac{a + 6 - a \cdot 2 - a (- a) - 3 \cdot 2 - 3 (- a)}{(a + 2) (2 - a)} + \frac{a - 3}{2 - a}$

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $- 1$ .

$\frac{a + 6 - 2 a - a (- a) - 3 \cdot 2 - 3 (- a)}{(a + 2) (2 - a)} + \frac{a - 3}{2 - a}$

Rewrite using the commutative property of multiplication.

$\frac{a + 6 - 2 a - 1 \cdot - 1 a \cdot a - 3 \cdot 2 - 3 (- a)}{(a + 2) (2 - a)} + \frac{a - 3}{2 - a}$

Multiply $a$ by $a$ by adding the exponents.

Move $a$ .

$\frac{a + 6 - 2 a - 1 \cdot - 1 (a \cdot a) - 3 \cdot 2 - 3 (- a)}{(a + 2) (2 - a)} + \frac{a - 3}{2 - a}$

Multiply $a$ by $a$ .

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

Multiply $- 1$ by $- 1$ .

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

Multiply $a^{2}$ by $1$ .

$\frac{a + 6 - 2 a + a^{2} - 3 \cdot 2 - 3 (- a)}{(a + 2) (2 - a)} + \frac{a - 3}{2 - a}$

Multiply $- 3$ by $2$ .

$\frac{a + 6 - 2 a + a^{2} - 6 - 3 (- a)}{(a + 2) (2 - a)} + \frac{a - 3}{2 - a}$

Multiply $- 1$ by $- 3$ .

$\frac{a + 6 - 2 a + a^{2} - 6 + 3 a}{(a + 2) (2 - a)} + \frac{a - 3}{2 - a}$

Add $- 2 a$ and $3 a$ .

$\frac{a + 6 + a + a^{2} - 6}{(a + 2) (2 - a)} + \frac{a - 3}{2 - a}$

Add $a$ and $a$ .

$\frac{2 a + 6 + a^{2} - 6}{(a + 2) (2 - a)} + \frac{a - 3}{2 - a}$

Subtract $6$ from $6$ .

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

Add $2 a + a^{2}$ and $0$ .

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

Factor $a$ out of $2 a + a^{2}$ .

Factor $a$ out of $2 a$ .

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

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

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

Factor $a$ out of $a \cdot 2 + a \cdot a$ .

$\frac{a (2 + a)}{(a + 2) (2 - a)} + \frac{a - 3}{2 - a}$

Cancel the common factor of $2 + a$ and $a + 2$ .

Reorder terms.

$\frac{a (a + 2)}{(a + 2) (2 - a)} + \frac{a - 3}{2 - a}$

Cancel the common factor.

$\frac{a (a + 2)}{(a + 2) (2 - a)} + \frac{a - 3}{2 - a}$

Rewrite the expression.

$\frac{a}{2 - a} + \frac{a - 3}{2 - a}$

Step 6

Simplify terms.

Combine the numerators over the common denominator.

$\frac{a + a - 3}{- a + 2}$

Add $a$ and $a$ .

$\frac{2 a - 3}{- a + 2}$

Factor $- 1$ out of $- a$ .

$\frac{2 a - 3}{- (a) + 2}$

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

$\frac{2 a - 3}{- (a) - 1 (- 2)}$

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

$\frac{2 a - 3}{- (a - 2)}$

Rewrite negatives.

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

$\frac{2 a - 3}{- 1 (a - 2)}$

Move the negative in front of the fraction.

$- \frac{2 a - 3}{a - 2}$

Simplify (a+6)/(4-a^2)-(a+3)/(a+2)+(a-3)/(2-a)

Related Questions

Support LCMGCF.com by disabling your adblocker.