1. If k=11, m = 24, we have the equation x^2+11x-24 = 0 that doesn't satisfy both roots are negative.
2. " in order to satisfy Statement (1), one of the roots is positive and one is negative...": the given equation can also have no root or one root with k=5.
Abhishekgmat87 wrote:
Official Solution -
Evaluate Statement (1). Since there are variables, Plug In. If k = 5 and x = 2, then the equation in the question stem is \(2^2\) + (5)(2) – m = 0 which simplifies to 4 + 10 – m = 0 or 14 – m = 0 and m = 14. However, if k = 5 and x = 6, then the equation is 62 + (5)(6) – m = 0 which simplifies to 36 + 30 – m = 0 or 66 – m = 0 and m = 66. When different numbers that satisfy a statement yield different answers to the question, the statement is insufficient. So, write down BCE.
Now, evaluate Statement (2). It states that if both roots are negative, then k = 11. If the two roots are negative, then the two factors are positive. Since k is the sum of the last terms in the two factors and m is the product, the factors could be (x + 2)(x + 9). In that case, m = 2 × 9 =18. However, if the two factors are (x + 3)(x + 8), then m = 3 × 8 = 24. When different numbers that satisfy a statement yield different answers to the question, the statement is insufficient. Eliminate choice B.
Now, evaluate both statements together. Statement (2) states that if both roots are negative, then k = 11 and Statement (1) states that k = 5. So, in order to satisfy Statement (1), one of the roots is positive and one is negative because if both factors were positive, then the last terms would have to be negative. This means that the sum of the absolute value of the two last terms is 11 and the difference is 5. The only values that satisfy both statements are the factors (x + 8)(x – 3). Therefore, the original equation is \(x^2\) + 5x – 24 = 0 and m = 24. Since both statements together provide one specific answer to the question, which is 24, the statements together are sufficient. Eliminate choice E.
The correct answer is choice C.