Question

What is the value of \(x\)?
Statement I: \(2^2 + 5x + 6 = 0\)
Statement II: \(2^2 + 7x + 12 = 0\)

• A. Statement I alone is sufficient
• B. Statement II alone is sufficient
• C. Both statements together are sufficient
• D. Neither statement is sufficient

Hint:

When solving for a variable using statements, check if both statements lead to a single value or if they provide different solutions. If they do, neither alone may be sufficient.

Answer 1

The Correct Option is D

We will evaluate each statement to find the value of \(x\) and check if we can conclude a unique answer. Statement I: \[ 2^2 + 5x + 6 = 0 \quad \Rightarrow \quad 4 + 5x + 6 = 0 \quad \Rightarrow \quad 5x + 10 = 0 \quad \Rightarrow \quad 5x = -10 \quad \Rightarrow \quad x = -2 \] Statement II: \[ 2^2 + 7x + 12 = 0 \quad \Rightarrow \quad 4 + 7x + 12 = 0 \quad \Rightarrow \quad 7x + 16 = 0 \quad \Rightarrow \quad 7x = -16 \quad \Rightarrow \quad x = -\frac{16}{7} \] The two statements provide different values for \(x\): - From Statement I, \(x = -2\) - From Statement II, \(x = -\frac{16}{7}\) Since each statement gives a different value for \(x\), neither statement alone can be considered sufficient to determine a unique value for \(x\). Therefore, both statements together do not provide one definitive solution. Thus, the correct answer is: \[ \boxed{(D) Neither statement is sufficient} \]