Average of P and its reciprocal = \(\frac{\frac{1}{P} + P}{2}\)

\(=\frac{\frac{1 + P^2}{P}}{\frac{2}{1}}\)

\(=\frac{1+P^2}{2P}\)

So, we have:
QUANTITY A: \(\frac{1+P^2}{2P}\)

QUANTITY B: \(1\)

We can complete this solution using matching operations

Since \(2P\) is POSITIVE, we can safely multiply both quantities by \(2P\) to get:
QUANTITY A: \(1+P^2\)
QUANTITY B: \(2P\)

If you notice that we have a PERFECT SQUARE hiding among the two quantities, then we can first subtract \(2P\) from both quantities to get:
QUANTITY A: \(1+P^2-2P\)
QUANTITY B: \(0\)

Rearrange to get:
QUANTITY A: \(P^2-2P+1\)
QUANTITY B: \(0\)

Factor to get:
QUANTITY A: \((P-1)^2\)
QUANTITY B: \(0\)

Since we know that P-1 does not equal 0, we know that \((P-1)^2\) is POSITIVE

Answer: A

RELATED VIDEO