Question

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

Answer 1

Step 1:

Form a quadratic equation:

Let the consecutive even integers be $2x$ and $2x+2$

According to the question,

$\begin{array}{rcl}{\left(2x\right)}^2+{\left(2x+2\right)}^2& =& 340\\ & \Rightarrow & 4x^2+4x^2+8x+4=340\\ & \Rightarrow & 8x^2+8x+4-340=0\\ & \Rightarrow & 8x^2+8x-336=0\\ & \Rightarrow & x^2+x-42=0\end{array}$

Step 2:

Determine the consecutive even numbers:

Use the factorization method to find the roots of the obtained quadratic equation.

$\begin{array}{rcl}& \Rightarrow & x^2+7x-6x-42=0\\ & \Rightarrow & x\left(x+7\right)-6\left(x+7\right)=0\\ & \Rightarrow & \left(x-6\right)\left(x+7\right)=0\\ & \Rightarrow & x-6=0\mathrm{or}x+7=0\\ & \Rightarrow & x=6\mathrm{or}x=-7\end{array}$

Neglect the negative integer, $x=-7$.

$\therefore x=6$

So, the integers are $6\times 2=12$ and $6\times 2+2=14$

Hence, the required numbers are $12\mathrm{and}14$.