Question

If the sum of the squares of two consecutive even numbers is 244, what is the sum of the numbers?

• A. 22

Answer 1

The correct option is A 22
Let the required numbers be $x$ and $x+2.$
Then by question, we have

$x^2+(x+2{)}^2=244.$
$\Rightarrow x^2+x^2+4x+4=244$
$\Rightarrow 2x^2+4x-240=0$
$\Rightarrow x^2+2x-120=0$
Since the roots of a quadratic equation $a{x}^2+bx+c=0,$ where $a,b$ and $c$ are constants ($a\ne 0$) are given by
$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a},$ for the above equation we have
$x=\frac{-2\pm \sqrt{2^2-4\left(1\right)(-120)}}{2\times 1}.$
$⟹x=\frac{-2\pm \sqrt{484}}{2}$
$⟹x=\frac{-2\pm 22}{2}$
$⟹x=10,-12$
Since $x$ is even, we have $x=10$ and $x+2=10+2=12.$
Thus, the sum of the numbers is $10+12=22.$