Simplify (3y)/(y^2+3y-10)-5/(2y+10)

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

Simplify each term.

Factor $y^{2} + 3 y - 10$ 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 $- 10$ and whose sum is $3$ .

$- 2, 5$

Write the factored form using these integers.

$\frac{3 y}{(y - 2) (y + 5)} - \frac{5}{2 y + 10}$

Factor $2$ out of $2 y + 10$ .

Factor $2$ out of $2 y$ .

$\frac{3 y}{(y - 2) (y + 5)} - \frac{5}{2 (y) + 10}$

Factor $2$ out of $10$ .

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

Factor $2$ out of $2 y + 2 \cdot 5$ .

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

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

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

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

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

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

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

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

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

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

Reorder the factors of $(y - 2) (y + 5) \cdot 2$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $2$ by $3$ .

$\frac{6 y - 5 (y - 2)}{2 (y + 5) (y - 2)}$

Apply the distributive property.

$\frac{6 y - 5 y - 5 \cdot - 2}{2 (y + 5) (y - 2)}$

Multiply $- 5$ by $- 2$ .

$\frac{6 y - 5 y + 10}{2 (y + 5) (y - 2)}$

Subtract $5 y$ from $6 y$ .

$\frac{y + 10}{2 (y + 5) (y - 2)}$

Simplify (3y)/(y^2+3y-10)-5/(2y+10)

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