2025 · Q28
If p = (√5-2)/(√5+2) and q = (√5+2)/(√5-2), then what is the value of p/q + q/p?
Elementary MathematicsAlgebra
- A18
- B324
- C322
- D320
Explanation
q=1/p (their product is 1). So p/q+q/p=p^2+1/p^2. Rationalizing, p=(sqrt(5)-2)/(sqrt(5)+2)=(sqrt(5)-2)^2=9-4*sqrt(5), 1/p=9+4*sqrt(5). Then p^2+1/p^2=(9-4*sqrt(5))^2+(9+4*sqrt(5))^2=2(81+80)=322. Spotting q=1/p halves the work.