Question

A student solves the following equation and determines that the solution is −2. Is the student correct? Explain.

3
a + 2
−
6a
a2 − 4
=
1
a − 2

Answer 1

No — the student is not correct.

Start with the equation
3/(a+2) − 6a/(a^2−4) = 1/(a−2).

Note a^2 − 4 = (a−2)(a+2), so the expression is undefined at a = 2 and a = −2; these must be excluded from any solution.

Multiply both sides by (a−2)(a+2) (valid only for a ≠ ±2) to clear denominators:
3(a−2) − 6a = a+2.
Simplify: 3a − 6 − 6a = a + 2 ⇒ −3a − 6 = a + 2 ⇒ −4a = 8 ⇒ a = −2.

That value solves the cleared equation but is not allowed in the original because it makes the denominator a+2 equal to 0. Hence it is extraneous. Therefore the original equation has no solution.