A rectangle has area 60 and perimeter 34. What is the length of its diagonal?

You need not find the two sides at all — one identity carries you straight to the diagonal. Let the sides be l and w. The perimeter gives l + w = 34 ÷ 2 = 17, and the area gives l · w = 60. The diagonal, by the Pythagorean theorem, is √(l² + w²). Now use the identity (l + w)² = l² + 2lw + w²: so l² + w² = (l + w)² − 2lw = 17² − 2 · 60 = 289 − 120 = 169. The diagonal is √169 = 13. Takeaway: when you know a sum and a product, the sum of squares comes free from (sum)² − 2·(product) — no need to solve for the pieces.