Let a be largest real value of $x$ for which $x^3 - 8x^2 - 2x + 3 = 0.$ Let $a$ be largest real value of $x$ for which $x^3 - 8x^2 - 2x + 3 = 0$. Determine the integer closest to $a^2$.
How I tried to do this:
This is a third-degree polynomial, thus there are 3 positive/negative values of $x$. If I find the roots of this polynomial and take the largest one and square it, I should get $a^2$. I don't get the question's logic of "closest to $a^2$", why would I want to find that? How would I be able to find an integer "closest" to $a^2$? You either find it or you don't. I'd begin with testing small values, namely $0, -1, 1, 2, -2,$ etc. and then trying to find something that yields me an answer close to $0$. I'd then go on from there...
Thanks, would really appreciate help w/ this problem!

If $a$ is a root of $x^3 -8x^2-2x+3$ then $a^2$ is a root of $x^3 - 68x^2+52x-9$. It's not hard to see that this is negative at 67 and positive at 68, so the root is somewhere in between those. Then testing 67.5 we can also see it's positive so it's closer to 67. 
