Yesterday, Manhattan GMAT posted a GMAT question on our blog. Today, they have followed up with the answer:
For the given equation to have exactly one root, the quadratic expression on the left must factor into (x + k)2, so that the equation becomes (x + k)2 = 0. The single root of this equation would be x = –k.
Expand (x + k)2. You get x2 + 2xk + k2. Now match that up to the given expression x2 + ax + b. For these expressions to match, the coefficients of the x terms must be the same, and the constants must be the same as well.
This means that 2xk = ax and that k2 = b. Solve the first equation for k in terms of a. You get k = a/2. Plug into the second equation. You get a2/4 = b. This is the answer.
You can also test numbers to solve this problem. For instance, make up an a and a b so that the equation has one root.
x2 + 6x + 9 = 0 has just one root, because the left side factors to (x + 3)2. The solution is x = –3.
Likewise, x2 + 8x + 16 = 0 has just one root, because the left side factors to (x + 4)2. The solution is x = –4.
In the first case, a = 6 and b = 9. In the second case, a = 8 and b = 16. Only answer choice E works for both cases.
The correct answer is E.