Quadratic equation:

The difference between two positive numbers is 7 and the sum of their square is 289. Find the two numbers

Can you tell me how this works?

A graphing calculator finds the solution fairly quickly.

The two numbers are 8 and 15.

For those of you who do not have a graphing calculator, or where graphing calculators are not allowed in exams, here's how we can solve the problem.

Let the smaller of the positive numbers be p, and the other must be (p+7) since the difference is 7.

The sum of squares equals 289 can be expressed as

p^2 + (p+7) ^2=289

expand

p^2+7^2+14p+p^2=289

2p^2+14p-240=0

p^2+7p-120=0

Factor and solve

(p-8) (p+15) = 0

which means p-8=0 or p+15=0 by the zero product theorem giving solutions of p=8 or p=-15 (rejected because p must be a positive number)

So we conclude that the two numbers are 8 and 8+7=15.