James was solving the following problem and came up with no solution as an answer. His answer is incorrect. Find his mistake, tell how to correct it, and give the right answer to the problem.

In a certain area of the city, traffic increases or decreases depending on the time of day based on the formula \(x^2-12x-6=y\). Meanwhile the number of traffic tickets given at time (x) is given as y = 2x + 1. The value of y when both graphs intersect represents the number of tickets given out. What is the largest of these values?

James' work:

x² - 12x - 6 = 2x + 1

x² -14x - 7 = 0

Plug into the quadratic formula using a = 1, b = -14, c = -7

\(x=\frac{-(-14)\pm \sqrt{{-14}^2-4(1)(-7)}}{2(1)}\)

\(x=\frac{14\pm \sqrt{-168}}{2}\)

This problem has no real solution because the discriminant

\(-{14}^2-4\left(1\right)\left(-7\right)=-168\). When you get a negative number for the discriminant there is no real answer to the problem.