Question

Solve for x:
1/(a+b+x) = 1/a +1/b +1/c , [a,b and x are not equal to zero and x is not equal -(a+b)]

Answer 1

Answer:

(-ab*(a+b+c))-b^2*(a+c)-abc/(ab+bc+ca)

Answer 2

$\frac{1}{a + b + x} = \frac{1}{a} + \frac{1}{b} + \frac{1}{x} \\ \frac{1}{a + b + x} - \frac{1}{x} = \frac{1}{a} + \frac{1}{b} \\ \frac{x - (a + b + x)}{ax + bx + {x}^{2} } = \frac{a + b}{ab} \\ \frac{x - a - b - x}{ax + bx + {x}^{2} } = \frac{a + b}{ab} \\ \frac{ - (a + b)}{(a+ b)x + {x}^{2} } = \frac{(a + b)}{ab} \\ \frac{ - 1}{(a + b)x + {x}^{2} } = \frac{1}{ab} \\ - ab = {x}^{2} + (a + b)x \\ {x }^{2} + (a + b)x + ab = 0 \\ (x + a)(x + b) = 0 \\ \\ x + a = 0 \: \: \: \: \: or \: \: \: \: \: x + b = 0 \\ x = - a \: \: \: \: \: \: \: \: \: or \: \: \: \: \: \: \: \: \: \: \: \: x = - b$