Question

Find all values of $x$ such that
\[\frac{x}{x - 5} = \frac{4}{x - 4} + 2.\]

Answer 1

Solve
x/(x-5) = 4/(x-4) + 2, with x ≠ 4,5.

Rewrite RHS: 4/(x-4)+2 = (4 + 2(x-4))/(x-4) = 2(x-2)/(x-4). So
x/(x-5) = 2(x-2)/(x-4).

Cross-multiply:
x(x-4) = 2(x-2)(x-5)
x^2 - 4x = 2x^2 - 14x + 20
0 = x^2 - 10x + 20.

Solve quadratic:
x = [10 ± √(100 - 80)]/2 = (10 ± 2√5)/2 = 5 ± √5.

Both are allowed (neither is 4 or 5), so the solutions are x = 5 + √5 and x = 5 - √5.