Question

Find two consecutive positive even integers, the sum of whose squares is 340.

Answer 1

$$\begin{gathered} \text{Let the two consecutive}\mathrm{even\; positive}\text{integers be 2}x\text{and (2}x+2). \\ \text{According to the question:} \\ \text{(2}x{)}^2+{(2x+2)}^2=340 \\ \Rightarrow 4x^2+{\left(2x\right)}^2+2\times 2x\times 2+{\left(2\right)}^2=340 \\ \Rightarrow 4x^2+4x^2+8x+4=340 \\ \Rightarrow 8x^2+8x-336=0 \\ \Rightarrow 8(x^2+x-42)=0 \\ \Rightarrow x^2+x-42=0 \\ \Rightarrow x^2+(7-6)x-42=0 \\ \Rightarrow x^2+7x-6x-42=0 \\ \Rightarrow x(x+7)-6(x+7)=0 \\ \Rightarrow (x+7\left)\right(x-6)=0 \\ \Rightarrow x+7=0\text{or}x-6=0 \\ \Rightarrow x=-7\text{or}x=6 \\ \Rightarrow x=6(\because x\text{cannot be negative}) \\ \text{Hence, the two consecutive numbers are}\left\{\right(2\times 6)=12\}\text{and {}(2\times 6+2)=14\}. \end{gathered}$$