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rss GMAT Practice Question (One of Hundreds)
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?
  1. j = k
  2. k is negative
Correct Answer: B
  1. 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
  2. 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.
  3. 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.
  4. 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.
  5. 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.
  6. Since Statement (1) alone is NOT SUFFICIENT and Statement (2) alone is SUFFICIENT, answer B is correct.
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