Find two consecutive odd positive integers, the sum of whose squares is 394.

1. 1 · Write what is given and what is to be found. Let the integers be x and x + 2, where x is odd and positive. Sum of squares = 394.
2. 2 · Write the equation. x² + (x + 2)² = 394
3. 3 · Write it in standard form. x² + x² + 4x + 4 = 394, so 2x² + 4x − 390 = 0. Dividing by 2: x² + 2x − 195 = 0.
4. 4 · Split the middle term. Product −195, sum 2: the numbers are 15 and −13. x² + 15x − 13x − 195 = x(x + 15) − 13(x + 15) = (x + 15)(x − 13) = 0.
5. 5 · Find and select the roots. x = 13 or x = −15. Reject −15, since the integers are positive.
6. 6 · Check and box the answer. 13² + 15² = 169 + 225 = 394. Therefore the integers are 13 and 15.

Scoring tip: Divide the whole equation by a common factor (here 2) before splitting. Smaller numbers mean fewer mistakes.