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Cubic Tidbit

Some people are so clever...and I am one who sits by the wayside and applauds.

Consider the cubic equation x3-4x2+3x-7 = 0. Does it have 1, 2, or 3 real roots?

Someone pointed out that the above cubic can be easily converted into the equation x3 = 4x2-3x+7. Then, looking at the "quadratic" right side, its discriminant is 9-(4x4x7) < 0.

Therefore, the quadratic graphically intersects the cubic y = x3 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!