Find "t" where x = x(0)+v(0)t+1/2at^2?

It's a true statement, but not helpful since you have t on both sides of the equation. So that won't help you find t.

This is a quadratic equation. In standard form,

(a/2) t^2 + v0*t + (x0 - x) = 0

I'm going to multiply everything by 2 so that the coefficient of t^2 is a, which is the standard letter used.

at^2 + 2v0*t + 2(x0 - x) = 0

Now it's a standard quadratic ax^2 + bx + c = 0 with b = 2v0 and c = 2(x0 - x). Use the quadratic formula.

t = [-b +- sqrt(b^2 - 4ac)] / 2a

It's OK to write subscripts that way. It makes it looks like "x at time 0" and "v at time 0" which is in fact the meaning. A more standard notation is to use the underscore: x_0, y_0. That means "subscript" in many computer typesetting systems, just as the carat (^) means "superscript".