I am to study the following equation for real solutions:
$$x^3 - 3x^2 + 4 = 0$$
I can see that $x = 2$ is a solution.
Then, using polynomial long division, I get the factor $x^2 - x - 2$.
Now, using the quadratic equation to solve this factor for solutions, I get:
$$b^2 - 4ac = 1^2 - (4 * 1 * 2) = -7$$
As I understand from my notes, this means $x^2 - x - 2 = 0$ has no real solution.
If this is correct, can I now rest happy that the only real solution to the original equation is $x = 2$?