Question

Solve the following quadratic equations by factorization:
(a/x-a)+(b/x-b)=2c/x-c

Answer 1

x = 0 or, $x = \frac{2ab - bc - ca}{a + b - 2c}$

Step-by-step explanation:

We have to solve the below equation by factorization.

The equation is

$\frac{a}{x - a} + \frac{b}{x - b} = \frac{2c}{x - c}$

⇒ $\frac{a}{x - a} + 1 + \frac{b}{x - b} + 1 = \frac{2c}{x -c} + 2$

⇒ $\frac{x}{x - a} + \frac{x}{x - b} = \frac{2x}{x - c}$

⇒ $x[x - b + x - a](x - c) = 2x(x - a)(x - b)$

⇒ x(2x - a - b)(x - c) = 2x(x - a)(x - b)

⇒ x[(2x² - ax - bx - 2cx + ac + bc) - (2x² - 2ax - 2bx + 2ab)] = 0

⇒ x(ax + bx - 2cx - 2ab + bc + ca) = 0

Therefore, x = 0 or, (ax + bx - 2cx - 2ab + bc + ca) = 0

⇒ x = 0 or, $x = \frac{2ab - bc - ca}{a + b - 2c}$ (Answer)