Subtract (3x)/(x^2+3x-10)-(2x)/(x^2+x-6)

$\frac{3 x}{x^{2} + 3 x - 10} - \frac{2 x}{x^{2} + x - 6}$

Simplify each term.

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Factor $x^{2} + 3 x - 10$ using the AC method.

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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.

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

Factor $x^{2} + x - 6$ using the AC method.

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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 $1$ .

$- 2, 3$

Write the factored form using these integers.

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

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

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

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

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

Write each expression with a common denominator of $(x - 2) (x + 5) (x + 3)$ , by multiplying each by an appropriate factor of $1$ .

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Multiply $\frac{3 x}{(x - 2) (x + 5)}$ and $\frac{x + 3}{x + 3}$ .

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

Multiply $\frac{2 x}{(x - 2) (x + 3)}$ and $\frac{x + 5}{x + 5}$ .

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

Reorder the factors of $(x - 2) (x + 5) (x + 3)$ .

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

Reorder the factors of $(x - 2) (x + 3) (x + 5)$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

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Factor $x$ out of $3 x (x + 3) - 2 x (x + 5)$ .

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Factor $x$ out of $3 x (x + 3)$ .

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

Factor $x$ out of $- 2 x (x + 5)$ .

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

Factor $x$ out of $x (3 (x + 3)) + x (- 2 (x + 5))$ .

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

Apply the distributive property.

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

Multiply $3$ by $3$ .

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

Apply the distributive property.

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

Multiply $- 2$ by $5$ .

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

Subtract $2 x$ from $3 x$ .

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

Subtract $10$ from $9$ .

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

Subtract (3x)/(x^2+3x-10)-(2x)/(x^2+x-6)

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