Question

Solve for x (in terms of a and b):
A/x-b + B/x-a =2

Answer 1

SOLUTION

$\frac{a}{x - b} + \frac{b}{x - a} = 2 \\ = > \frac{a(x - a) + b(x - b)}{(x - a)(x - b)} = 2 \\ = > ax - a {}^{2} + bx - b {}^{2} = 2x {}^{2} + 2ab - 2bx - 2ax \\ = > - 2x {}^{2} + ax + bx + 2ax + 2bx = a {}^{2} + b {}^{2} + 2ab \\ = > 3ax + 3bx - 2x {}^{2} = (a + b) {}^{2} \\ = > 2x {}^{2} - x(3a + 3b) + (a + b) {}^{2} = 0 \\ = > 2x {}^{2} -x (2(a + b) + (a + b)) + (a + b) {}^{2} = 0 \\ = > 2x {}^{2} - 2x(a + b) - x(a + b) + (a + b) {}^{2} = 0 \\ = > 2x(x - (a + b)) - (a + b)(x - (a + b)) = 0 \\ = > (2x - (a + b))(x - (a + b)) = 0 \\ = > (2x - a - b)(x - a - b) = 0 \\ \\ therefore..... \\ either \\ 2x - a - b = 0 \\ = > x = \frac{a + b}{2} \\ or..... \\ x - a - b = 0 \\ = > x = a + b$

Answer 2

AnswEr :

➨ $\large \bold{ \frac{a}{x - b} + \frac{b}{x - a} = 2}$

➨ $\large \bold{ \frac{a(x - a) + b(x - b)}{(x - b)(x - a)} = 2 }$

• By Cross Multiplication

➨ a(x - a) + b(x - b) = 2(x - b)(x - a)

➨ ax - a² + bx - b² = 2(x² - ax - bx + ab)

➨ ax - a² + bx - b² = 2x² - 2ax - 2bx + 2ab

➨ ax + 2ax + bx + 2bx = 2x² + 2ab + a² + b²

➨ 3ax + 3bx = 2x² + (a + b)²

➨ 3ax + 3bx - 2x² - (a + b)² = 0

➨ - 2x² + (3a + 3b)x - (a + b)² = 0

➨ - 2x² + [(2a + 2b) + (a + b)]x - (a + b)² = 0

➨ - 2x² + (2a + 2b)x + (a + b)x - (a + b)² = 0

➨ - 2x[x - (a + b)] + (a + b)[x - (a + b)] = 0

➨ [(a + b) - 2x][x - (a + b)] = 0

➨ [(a + b) - 2x] = 0 or, [x - (a + b)] = 0

➨ (a + b) = 2x or, (x - a - b) = 0

➨ x = $\huge\tt{\frac{(a+b)}{2}}$ or, x = (a + b)