Find two consecutive positive even numbers the sum of whose squares is 340?

Let the number be 'm' & 'm + 2'

Hence their squares are m^2 & (m+2)^2

There sum is

m^2 + (m+2)^2 = 340

Multiply out he brackets

m^2 + m^2 + 4m + 4 = 340 .

2m^2 + 4m + 4 = 340

Divide both sides by '2'

m^2 + 2m + 2 = 170

m^2 + 2m - 168 = 0

Factor

(m - 12)(m + 14) = 0

Hence m = 12 & m+ 2 = 14 The two consecutive even numbers.

12^2 = 144

14^2 = 196

144 + 196 = 340 As required.