Practice GMAT Data Sufficiency Question

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A: x² + 6x - 40 = 0
B: x² + kx + j = 0
Which is larger, the sum of the roots of equation A or the sum of the roots of equation B?

j = k
k is negative

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)	Statement (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Correct Answer: B

This problem combines the quadratic formula with properties of positive and negative numbers. First, find the sum of the roots in equation A using the quadratic formula or factoring.
Using the quadratic formula:
x = (-6 + sqrt(36 - 4(1)(-40))) / 2 and (-6 - sqrt(36 - 4(1)(-40))) / 2 are the roots.

Using factoring:
x² + 6x - 40 = 0
(x + 10)(x - 4) = 0
x = -10, 4
To find the sum, these two roots will be added. Notice that one root contains +sqrt(36 + 160) and the other contains -sqrt(36 + 160). When these two terms are added, they equal zero. Thus, the only terms left in the sum are -6/2 and -6/2. Add these together to find the sum of the roots: -6/2 + (-6/2) = -6. Notice that the sum of the roots equals -b, where b is the coefficient of the x term.
In fact, in any sum of quadratic roots, the +sqrt(...) and -sqrt(...) terms will cancel. Therefore, for any quadratic equation the sum of the roots is -b, where b is the coefficient of the x term (ax² + bx + c = 0). This fact will simplify the problem greatly.
Evaluate Statement (1) alone.
1. The sum of the roots for equation A was found to be -6. Using the fact demonstrated above, the sum of the roots of equation B is -k. Statement (1) says that that j = k, which means that the sum of the roots of equation B is -k = -j.
2. However, nothing is known about j and k. It could be that j = -7, in which case the sum of the roots of B is -(-7) = 7, which is larger than the sum of the roots of A. However, it could be that j = 9, in which case the sum of the roots of B is -(9) = -9, which is smaller than the sum of the roots of A. It cannot be determined which sum is larger.
3. Note: We cannot assume that j and k are integers as the problem does not state this. If we knew they were integers, then j = k = 2 since this is the only way for j to equal k in x² + jx + k = 0, and we could solve the problem.
4. Statement (1) is NOT SUFFICIENT.
Evaluate Statement (2) alone.
1. If k is negative, then the sum of the roots of B is -k, which is the negative of a negative number, making the sum positive. And since this sum is positive, it is larger than the sum of the roots of A, which is -6.
2. Statement (2) is SUFFICIENT.
Since Statement (1) alone is NOT SUFFICIENT and Statement (2) alone is SUFFICIENT, answer B is correct.

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