Question

prove that 1/A+B+X=1/A+1/B+1/X​

Answer 1

$\frac{1}{a + b + x} = \frac{1}{a} + \frac{1}{b} + \frac{1}{x}$

$\frac{1}{a + b + x} - \frac{1}{x} = \frac{1}{a} + \frac{1}{b}$

$\frac{x - a - b - x}{ax + bx + x {}^{2} } = \frac{b + a}{ab}$

$\frac{ - a - b}{x {}^{2} + ax + bx } = \frac{a + b}{ab}$

$- (a + b)(ab) = (a + b)(x {}^{2} + ax + bx)$

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

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

$x(x + a) \: \: b(x + a) = 0 \\ (x + b) \: \: (x + a) = 0 \\ x = - b \\ x = - a$