One positive integer is 6 less than twice another. the sum of their squares is 680 . find the integers.

Let's call the two numbers x and y. The first sentence, one is 6 less than twice another, translates to

We also know the sum of their squares:

So, we have the following system:

We can use the expression for x in terms of y from the first equation to turn the second equation into something involving y alone:

Expand the square:

Subtract 680 from both sides:

Using the quadratic equation

we find the two solutions

Since we know that both numbers are integer we can only accept the second solution. It yields the following x value:

5.5

step-by-step explanation: