Question

Solve for all values of x.
x, plus, start fraction, 2, x, divided by, x, minus, 1, end fraction, equals, start fraction, 2, divided by, x, minus, 1, end fraction
x+
x−1
2x
​
=
x−1
2

Answer 1

Interpret the equation as
x + (2x)/(x − 1) = (x − 1)/2,
with x ≠ 1.

Multiply both sides by 2(x − 1) to clear denominators:
2(x − 1)x + 4x = (x − 1)^2.

Expand and simplify:
2x^2 − 2x + 4x = x^2 − 2x + 1
=> 2x^2 + 2x = x^2 − 2x + 1
=> x^2 + 4x − 1 = 0.

Solve the quadratic:
x = [−4 ± sqrt(16 + 4)]/2 = [−4 ± 2√5]/2 = −2 ± √5.

Both values are allowed (neither equals 1), so the solutions are
x = −2 + √5 and x = −2 − √5.