GRE quant question

Question

If $a > 0$ and $b < 0$, which of the following statements are true about the values of $x$ that solve the equation $x^2-ax+ b =0$?

Indicate all such statements.

A) They have opposite signs

B) Their sum is greater than zero.

C) Their product equals $-b$.

While solving some GRE practice problems, I came across the problem above. the question seemed easy enough at first sight, but after solving it and looking at the solution. the solution states that only choices A and B are correct. why is this true? why is choice C incorrect? I googled the question but I was not convinced by any of the answers. while glancing at the question again I also realized that the first statement seems to contradict the second. a>0 but x^2 - ax + b =0. how can "a" be positive but at the same time be negative in the quad. equation? "a" can be produced by the sum of the two roots of the equation. and how is choice C incorrect? the product of a positive and negative integer will produce a negative integer? I've been looking at this question for about 30 mins and i cannot help but feel that this is a poorly written gottca question. A good explanation is appreciated thanks.

Answer 1

The roots have sum $a>0$ (so B) is true) and product $b<0$ (so A) is true and C) is false).

Answer 2

Here is a concrete example of how this might come about.

Suppose $a = 2$. This satisfies the condition $a > 0.$

Suppose $b = -3.$ This satisfies the condition $b < 0.$ It also implies that $-b = -(-3) = 3.$

Now consider the equation $x^2-ax+ b =0$. Making the substitutions for $a$ and $b$ with the values we have supposed, the equation is literally

$$ x^2 - 2x - 3 = 0. $$

Now $a$ is still $2,$ and $2$ is still a positive number. Of course that also means $-2$, which is the coefficient of $x$ in this equation, is a negative number. But the only way $-2$ (or $-a$) could be negative is if $2$ (or $a$) is positive. The statement "$a$ is positive" doesn't contradict the statement "$-a$ is negative"; in order for the second statement to be true, the first statement has to be true.

The roots of this equation are $-1, 3.$ The have opposite signs and their sum is greater than zero. But the product of the roots is $-1 \times 3 = -3,$ whereas $-b$ is $3.$ Without doubt, $-3 \neq 3.$

Note: This is not a proof of the correct responses to the GRE. Look in the other answer of this question to find a proof of those responses. This answer is a disproof of the mistaken notions in the question.