Question

What is true about the solution of StartFraction x squared Over 2 x minus 6 EndFraction = StartFraction 9 Over 6 x minus 18 EndFraction?
x = plus-or-minus StartRoot 3 EndRoot, and they are actual solutions.
x = plus-or-minus StartRoot 3 EndRoot, but they are extraneous solutions.
x = 3, and it is an actual solution.
x = - 3, but it is an extraneous solution.

Answer 1

x = ±√3, and they are actual solutions.

Reason: 2x−6 = 2(x−3) and 6x−18 = 6(x−3). For x ≠ 3 the equation reduces to x^2/(2(x−3)) = 3/(2(x−3)) ⇒ x^2 = 3, so x = ±√3. Neither ±√3 makes a denominator zero, so both are valid solutions.