Browse · harp

smc

algebra intermediate

Problem

The expression

\frac{P + Q}{P - Q} - \frac{P - Q}{P + Q}

where

P = x + y

and

Q = x - y

, is equivalent to:

(A)

\frac{x ^{2} - y ^{2}}{x y}

(B)

\frac{x ^{2} - y ^{2}}{2 x y}

(C)

1

(D)

\frac{x ^{2} + y ^{2}}{x y}

Solution

Simplifying before substituting by obtaining a common denominator gives:

\frac{( P + Q ) ^{2} - ( P - Q ) ^{2}}{( P - Q ) ( P + Q )}

\frac{( P ^{2} + 2 P Q + Q ^{2} ) - ( P ^{2} - 2 P Q + Q ^{2} )}{P ^{2} - Q ^{2}}

\frac{4 P Q}{P ^{2} - Q ^{2}}

Substituting gives:

\frac{4 ( x + y ) ( x - y )}{( x + y ) ^{2} - ( x - y ) ^{2}}

\frac{4 ( x ^{2} - y ^{2} )}{( x ^{2} + 2 x y + y ^{2} ) - ( x ^{2} - 2 x y + y ^{2} )}

\frac{4 ( x ^{2} - y ^{2} )}{4 x y}

\frac{x ^{2} - y ^{2}}{x y}

\frac{x ^{2} - y ^{2}}{x y}

Final answer

A