Cubic Tidbit
Some people are so clever...and I am one who sits by the wayside and applauds.
Consider the cubic equation x^{3}4x^{2}+3x7 = 0. Does it have 1, 2, or 3 real roots?
Someone pointed out that the above cubic can be easily converted into the equation x^{3} = 4x^{2}3x+7. Then, looking at the "quadratic" right side, its discriminant is 9(4x4x7) < 0.
Therefore, the quadratic graphically intersects the cubic y = x^{3} only once...which means that the original cubic has only one real root.
Clever...and quite easy to do....if one cannot find their graphing calculator!
