perform the indicated operations: $\frac{2}{x - 2}+\frac{x}{x + 9}-\frac{x + 20}{x^{2}+7x…

perform the indicated operations: $\frac{2}{x - 2}+\frac{x}{x + 9}-\frac{x + 20}{x^{2}+7x - 18}$\n$\frac{x^{2}-x + 2}{x^{2}+7x - 18}$\n$\frac{x + 1}{x + 9}$\n$\frac{22}{x^{2}+7x - 18}$\n$\frac{x^{2}+x + 38}{x^{2}+7x - 18}$

perform the indicated operations: $\frac{2}{x - 2}+\frac{x}{x + 9}-\frac{x + 20}{x^{2}+7x - 18}$\n$\frac{x^{2}-x + 2}{x^{2}+7x - 18}$\n$\frac{x + 1}{x + 9}$\n$\frac{22}{x^{2}+7x - 18}$\n$\frac{x^{2}+x + 38}{x^{2}+7x - 18}$

Answer

Explanation:

Step1: Factor the denominator

Factor $x^{2}+7x - 18=(x - 2)(x+9)$.

Step2: Find a common - denominator

The common denominator of $\frac{2}{x - 2},\frac{x}{x + 9},\frac{x + 20}{x^{2}+7x - 18}$ is $(x - 2)(x + 9)$. Rewrite the fractions: $\frac{2}{x - 2}=\frac{2(x + 9)}{(x - 2)(x + 9)}=\frac{2x+18}{(x - 2)(x + 9)}$, $\frac{x}{x + 9}=\frac{x(x - 2)}{(x - 2)(x + 9)}=\frac{x^{2}-2x}{(x - 2)(x + 9)}$, and $\frac{x + 20}{x^{2}+7x - 18}=\frac{x + 20}{(x - 2)(x + 9)}$.

Step3: Combine the fractions

$\frac{2x+18}{(x - 2)(x + 9)}+\frac{x^{2}-2x}{(x - 2)(x + 9)}-\frac{x + 20}{(x - 2)(x + 9)}=\frac{(2x + 18)+(x^{2}-2x)-(x + 20)}{(x - 2)(x + 9)}$.

Step4: Simplify the numerator

$(2x+18)+(x^{2}-2x)-(x + 20)=2x+18+x^{2}-2x - x - 20=x^{2}-x - 2$. So the result is $\frac{x^{2}-x - 2}{x^{2}+7x - 18}$.

Answer:

$\frac{x^{2}-x - 2}{x^{2}+7x - 18}$