Question

Find two consecutive odd positive integers, sum of whose squares is 290.

Answer 1

Let the smaller of the two consecutive odd positive integers be x. Then, the second integer will be x + 2. According to the question,

x² + (x + 2)² = 290

i.e., x² + x² + 4x + 4 = 290

i.e., 2x² + 4x – 286 = 0

i.e., x² + 2x – 143 = 0

which is a quadratic equation in x.

Using the quadratic formula, we get

But x is given to be an odd positive integer. Therefore, x≠ – 13, x = 11.

Thus, the two consecutive odd integers are 11 and 13.

Check : 11² + 13² = 121 + 169 = 290.