Question

The sum of two integers is 26. the sum of the squares of the two integers is 340. what is the product of the two numbers?

Answer 1

Lets say that the two unknown integers are $n$ and $m$.

We know the following things about $n$ and $m$:

$n+m=26$
$n^2+m^2=340$

And, we want to find $nm$.

To solve this, we'll use the expansion of the squared of the sum of any two inegers; this is expressed as:

$(n+m)^2=n^2+2nm+m^2$

So, given what we know about the unknown integers, the previous can be written as:

$(26)^2=340+2nm$

We can easily solve for $nm$:

$nm= \frac{(26)^2-340}{2}= \frac{676-340}{2}= \frac{336}{2}=168$

The answer is 168.

Another approach to solve the problem is, from the two starting equations, compute the values of $n$ and $m$, which are 12 and 14, and directly compute their product; however, the approach described is more elegant.