Question

If $r$ and $s$ are the root of the equation $x^2+bx+c=0.$ where $b$ and $c$ are constants, is $rs<0$ ?
(1) $b<0$
(2) $c<0$

• A. Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
• C. Both statements together are sufficient, but neither statement alone is sufficient.
• D. Each statement alone is sufficient.
• E. Statements (1) and (2) together are not sufficient.

Answer 1

The correct option is B Statement (2) alone is sufficient, but statement (1) alone is not sufficient.

• Determine whether the product of the roots to , where b and c are constants, is negative. If r and s are the roots of the given equation, then (x − r)(x − s) = . This implies that , and so rs = c. Therefore, rs is negative if and only if c is negative.

1. Given that b < 0, then c could be negative or positive. For example, if b = −1 and c = −6, then the given equation would be , and the product of its roots would be (3)(−2), which is negative. On the other hand, if b = −6 and c = 5, then the given equation would be , and the product of its roots would be (5)(1), which is positive; NOT sufficient.
2. Given that c < 0, it follows from the explanation above that rs < 0; SUFFICIENT.

• The correct answer is B; statement 2 alone is sufficient.