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

Each statement will be analyzed to ascertain the value of \(x\) and determine if a unique solution can be derived. 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 values of \(x\) derived from the two statements differ: - Statement I yields \(x = -2\). - Statement II yields \(x = -\frac{16}{7}\). As each statement provides a distinct value for \(x\), neither statement individually is sufficient to establish a unique value for \(x\). Consequently, combining both statements does not yield a single definitive solution. Thus, the appropriate answer is: \[\boxed{(D) Neither statement is sufficient}\]