The hypotenuse of a right-angled triangle is 17 cm. One of the other two sides is 7 cm longer than the third side. Represent this as a quadratic equation and find the sides.

1. 1 · Write what is given and what is to be found. Hypotenuse = 17 cm. Let the shorter side be x cm; then the other side is (x + 7) cm.
2. 2 · Draw the figure. Draw a right triangle. Mark the right angle and label the sides x, x + 7 and 17 cm (the side opposite the right angle).
3. 3 · Write the theorem. By Pythagoras' theorem: x² + (x + 7)² = 17².
4. 4 · Form the equation. x² + x² + 14x + 49 = 289, so 2x² + 14x − 240 = 0, and dividing by 2: x² + 7x − 120 = 0.
5. 5 · Solve. x² + 15x − 8x − 120 = (x + 15)(x − 8) = 0, so x = 8 or x = −15. Reject −15: a length cannot be negative.
6. 6 · Check and box the answer. 8² + 15² = 64 + 225 = 289 = 17². Therefore the sides are 8 cm and 15 cm.

Scoring tip: Write 'rejected, since a length cannot be negative' next to the negative root. Examiners look for that reason.