Bunuel wrote:
\((x+\frac{1}{2})^2 - (2x-4)^2 =\)
(A) \(−3x^2 − 15x + \frac{65}{4}\)
(B) \(3x^2 + 16x\)
(C) \(−3x^2 + 17x − \frac{63}{4}\)
(D) \(5x^2 + \frac{65}{4}\)
(E) \(3x^2\)
One option here is to FOIL both parts and then combine terms. Another option is to treat the expression as a difference of squares.
However, I think the fastest approach is to test an easy value of \(x\).
Since we're looking for an expression that's
equivalent to \((x+\frac{1}{2})^2 - (2x-4)^2 \), let's first evaluate this expression for a certain value of \(x\), and then look for an answer choice that has the same value for that value of \(x\)
Let's see what happens when we plug \(x = 0\) into the given expression:
\((x+\frac{1}{2})^2 - (2x-4)^2 =(0+\frac{1}{2})^2 - (2(0)-4)^2=(\frac{1}{2})^2 - (-4)^2 =(\frac{1}{4}) - 16 = \frac{1}{4}-\frac{64}{4}=-\frac{63}{4}\)
So, the given expression evaluates to equal \(-\frac{63}{4}\) when \(x = 0\).
We can now plug \(x = 0\) into each answer choice to see which one evaluates to \(-\frac{63}{4}\)
(A) \(−3(0)^2 − 15(0) + \frac{65}{4}=\frac{65}{4}\). ELIMINATE
(B) \(3(0)^2 + 16(0)=0\). ELIMINATE
(C) \(−3(0)^2 + 17(0) − \frac{63}{4}=-\frac{63}{4}\). KEEP
(D) \(5(0)^2 + \frac{65}{4}=\frac{65}{4}\). ELIMINATE
(E) \(3(0)^2=0\). ELIMINATE
Answer: C