MathRevolution wrote:
[GMAT math practice question]
(Number Properties) \(x\) is an integer. What is the value of \(x\)?
1) \(x^2+4x+9\) is a perfect square.
2) \(x\) is a non-zero integer.
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Since we have \(1\) variable (\(x\)) and \(0\) equations, D is most likely the answer. So, we should consider each condition on its own first.
Condition 1)
We have \(x^2 + 4x + 9 = k^2\) for some integer \(k.\)
Then we have
\(k^2 – (x^2 + 4x + 4) = 5\)
\(k^2 – (x + 2)^2 = 5\)
or \((k + x + 2)(k – x - 2) = 5.\)
We have four cases
Case 1) \(k + x + 2 = 1, k – x – 2 = 5\)
Adding both equations together gives us:
\(k + x + 2 + k – x – 2 = 1 + 5\)
\(2k = 6\)
\(k = 3\)
Then \(k + x + 2 = 1\) becomes
\(3 + x + 2 = 1\)
\(x = -4\)
Then we have \(k = 3, x = -4.\)
Case 2) \(k + x + 2 = 5, k – x – 2 = 1\)
Adding both equations together gives us:
\(k + x + 2 + k – x – 2 = 5 + 1\)
\(2k = 6\)
\(k = 3\)
Then \(k + x + 2 = 5 \)becomes
\(3 + x + 2 = 5\)
\(x = 0\)
Then we have \(k = 3, x = 0.\)
Case 3) \(k + x + 2 = -1, k – x – 2 = -5\)
Adding both equations together gives us:
\(k + x + 2 + k – x – 2 = -1 + -5\)
\(2k = -6\)
\(k = -3\)
Then \(k + x + 2 = -1\) becomes
\(-3 + x + 2 = -1\)
\(x = 0\)
Then we have \(k = -3, x = 0.\)
Case 4) \(k + x + 2 = -5, k – x – 2 = -1\)
Adding both equations together gives us:
\(k + x + 2 + k - x - 2 = -5 + -1\)
\(2k = -6\)
\(k = -3\)
Then \(k + x + 2 = -5\) becomes
\(-3 + x + 2 = -5\)
\(x = -4\)
Then we have \(k = -3, x = -4.\)
Thus, we have two solutions for \(x\), which are \(0\) and \(-4\).
Since condition 1) does not yield a unique solution, it is not sufficient.
Condition 2)
Since condition 2) does not provide enough information to yield a unique solution, it is not sufficient.
Conditions 1) & 2)
We have a unique solution \(-4.\)
Since both conditions together yield a unique solution, they are sufficient.
Therefore, C is the answer.
Answer: C
If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations,” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A, B, C, or E.
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