06/07/2020 (2011: National Team Round, Problem 10)

Q: Three real numbers a, b, and c satisfy the equations: a + b + c = 2, ab + ac + bc = -7 and abc = -14. Express your answer in simplest radical form.

A: √7

So we know that the largest number is a radical, but none of the sum, product, and sum of products contain a radical, so that means that another number is a radical, but it's the negative (so as to 'eliminate' the radicals in ab + ac + bc).

If the radicals are a, and -a (=c), then that means ab + ac + bc = 7 becomes ab + a(-a) + (-a)b = 7, that simplifies to -a² = -7, which makes a² = 7, which means a = √7.