Answer
$\{-20,30\}$.
Work Step by Step
Determine the Least Common Denominator LCD $x(x+30)$ to clear fractions.
Multiply the equation by the LCD.
$x(x+30)\left ( \frac{40}{x}+\frac{40}{x+30}\right )=x(x+30)(2)$
Use the distributive property.
$x(x+30)\cdot \frac{40}{x}+x(x+30)\cdot \frac{40}{x+30}=x(x+30)(2)$
Cancel common factors.
$40(x+30)+40x=2x(x+30)$
Simplify.
$40x+1200+40x=2x^2+60x$
$80x+1200=2x^2+60x$
Add both sides $-80x-1200$.
$80x+1200-80x-1200=2x^2+60x-80x-1200$
Simplify.
$0=2x^2-20x-1200$
Divide both sides by $2$.
$0=x^2-10x-600$
Rewrite the middle term $-10x$ as $-30x+20x$.
$0=x^2-30x+20x-600$
Group terms.
$0=(x^2-30x)+(20x-600)$
Factor each term.
$0=x(x-30)+20(x-30)$
Factor out $(x-30)$.
$0=(x-30)(x+20)$.
Set each factor equal to zero.
$x-30=0$ or $x+20=0$
Isolate $x$.
$x=30$ or $x=-20$
The solution set is $\{-20,30\}$.
Note: We have to check if the solution is correct. The equation is defined for all real values of $x$ except the zeros of the denominators, which are $0$ and $-30$. Because none of the solutions we got ($x=30$ and $x=-20$) is zero of a denominators, they are correct.
