Simplify (2p-3)/(p^2-5p+6)-5/(p^2-9)

$\frac{2 p - 3}{p^{2} - 5 p + 6} - \frac{5}{p^{2} - 9}$

Simplify each term.

Factor $p^{2} - 5 p + 6$ 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 $6$ and whose sum is $- 5$ .

$- 3, - 2$

Write the factored form using these integers.

$\frac{2 p - 3}{(p - 3) (p - 2)} - \frac{5}{p^{2} - 9}$

Simplify the denominator.

Rewrite $9$ as $3^{2}$ .

$\frac{2 p - 3}{(p - 3) (p - 2)} - \frac{5}{p^{2} - 3^{2}}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = p$ and $b = 3$ .

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

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

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

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

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

Write each expression with a common denominator of $(p - 3) (p - 2) (p + 3)$ , by multiplying each by an appropriate factor of $1$ .

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

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

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

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

Reorder the factors of $(p - 3) (p - 2) (p + 3)$ .

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

Reorder the factors of $(p + 3) (p - 3) (p - 2)$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $p$ by $p$ by adding the exponents.

Move $p$ .

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

Multiply $p$ by $p$ .

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

Multiply $3$ by $2$ .

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

Multiply $- 3$ by $3$ .

$\frac{2 p^{2} + 6 p - 3 p - 9 - 5 (p - 2)}{(p + 3) (p - 2) (p - 3)}$

Subtract $3 p$ from $6 p$ .

$\frac{2 p^{2} + 3 p - 9 - 5 (p - 2)}{(p + 3) (p - 2) (p - 3)}$

Apply the distributive property.

$\frac{2 p^{2} + 3 p - 9 - 5 p - 5 \cdot - 2}{(p + 3) (p - 2) (p - 3)}$

Multiply $- 5$ by $- 2$ .

$\frac{2 p^{2} + 3 p - 9 - 5 p + 10}{(p + 3) (p - 2) (p - 3)}$

Subtract $5 p$ from $3 p$ .

$\frac{2 p^{2} - 2 p - 9 + 10}{(p + 3) (p - 2) (p - 3)}$

Add $- 9$ and $10$ .

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

Simplify (2p-3)/(p^2-5p+6)-5/(p^2-9)

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