What is the value of x?
- ABoth statements together are sufficient, but neither statement alone is sufficient.
- BStatements (1) and (2) together are not sufficient.
- CStatement (1) alone is sufficient, but statement (2) alone is not sufficient.
- DStatement (2) alone is sufficient, but statement (1) alone is not sufficient.
Show answer & explanationAnswer & explanation
Correct answer: A. Both statements together are sufficient, but neither statement alone is sufficient.
Statement (1): x^2 - 5x + 6 = 0. This factors to (x-2)(x-3) = 0, so x = 2 or x = 3. Not sufficient as x could be 2 or 3. Statement (2): x is a prime number. This tells us x could be 2, 3, 5, 7, etc. Not sufficient. Combining (1) and (2): From (1), x is 2 or 3. From (2), x is a prime number. Both 2 and 3 are prime numbers. So, x could still be 2 or 3. Therefore, both statements together are not sufficient to find a unique value for x.
Why the other options are wrong
- B. Correct. Both statements together yield x = 2 or x = 3. Since there isn't a unique value, the statements are not sufficient.
- C. Incorrect. Statement (1) yields two possible values for x.
- D. Incorrect. Statement (2) yields infinitely many possible values for x.
Quadratic Equations in Data Sufficiency
When a quadratic equation is given in Data Sufficiency, it often yields two possible values for the variable, requiring additional information to determine a unique solution.
- Factoring or quadratic formula gives solutions.
- Two distinct real roots are common.
- Additional statements must narrow down to a single root.
Memory trick: Unique means one, no more, no less; else, it fails the sufficiency test.