Quadratic Equations-45
From Homeworkwiki
Solve the following equation:
(x - a)/b + (x - b)/a = b/(x - a) + a/(x - b)
Solution: We have, (x - a)/b + (x - b)/a = b/(x - a) + a/(x - b)
=>(x - a)/b - b/(x - a) = a/(x - b) – (x - b)/a
=> [(x - a)2 - b2] / b(x - a) = [a2 - (x - b)2] / a(x - b)
=> (x – a - b)(x – a + b) / b(x - a) = (a – x + b)(a + x - b) / a(x - b)
=> (x – a - b)(x – a + b) / b(x - a) = - (x – a - b)(x + a - b) / a(x - b)
=> (x – a - b) [(x – a - b) / b(x - a) + (x + a - b) / a(x - b)] = 0
=> x – a - b = 0 or, [(x – a - b) / b(x - a) + (x + a - b) / a(x - b)] = 0
=> x = a + b, or, [(x – a - b) / b(x - a) + (x + a - b) / a(x - b)] = 0
Now, [(x – a - b) / b(x - a) + (x + a - b) / a(x - b)] = 0
=>[x – (a - b)]a(x - b) + b(b - x)(x + a - b) = 0
=> ax2 - ax(a - b) + ab(a - b) + bx2 + bx(a - b) – ab(a - b) = 0
=> x2(a + b) – x(a2 + b2) = 0
=> x[x(a + b) – (a2 + b2)] = 0
=> x = 0, or x = (a2 + b2) / (a + b)
Hence, the roots of the given equation are 0, a + b, (a2 + b2).
