Simplify m/(m-3)+(5m-48)/(m^2+5m-24)

$\frac{m}{m - 3} + \frac{5 m - 48}{m^{2} + 5 m - 24}$

Step 1

Factor $m^{2} + 5 m - 24$ 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 $- 24$ and whose sum is $5$ .

$- 3, 8$

Write the factored form using these integers.

$\frac{m}{m - 3} + \frac{5 m - 48}{(m - 3) (m + 8)}$

Step 2

To write $\frac{m}{m - 3}$ as a fraction with a common denominator, multiply by $\frac{m + 8}{m + 8}$ .

$\frac{m}{m - 3} \cdot \frac{m + 8}{m + 8} + \frac{5 m - 48}{(m - 3) (m + 8)}$

Step 3

Simplify terms.

Multiply $\frac{m}{m - 3}$ by $\frac{m + 8}{m + 8}$ .

$\frac{m (m + 8)}{(m - 3) (m + 8)} + \frac{5 m - 48}{(m - 3) (m + 8)}$

Combine the numerators over the common denominator.

$\frac{m (m + 8) + 5 m - 48}{(m - 3) (m + 8)}$

Step 4

Simplify the numerator.

Apply the distributive property.

$\frac{m \cdot m + m \cdot 8 + 5 m - 48}{(m - 3) (m + 8)}$

Multiply $m$ by $m$ .

$\frac{m^{2} + m \cdot 8 + 5 m - 48}{(m - 3) (m + 8)}$

Move $8$ to the left of $m$ .

$\frac{m^{2} + 8 \cdot m + 5 m - 48}{(m - 3) (m + 8)}$

Add $8 m$ and $5 m$ .

$\frac{m^{2} + 13 m - 48}{(m - 3) (m + 8)}$

Factor $m^{2} + 13 m - 48$ 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 $- 48$ and whose sum is $13$ .

$- 3, 16$

Write the factored form using these integers.

$\frac{(m - 3) (m + 16)}{(m - 3) (m + 8)}$

Step 5

Cancel the common factor of $m - 3$ .

Cancel the common factor.

$\frac{(m - 3) (m + 16)}{(m - 3) (m + 8)}$

Rewrite the expression.

$\frac{m + 16}{m + 8}$

Simplify m/(m-3)+(5m-48)/(m^2+5m-24)

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