1.8 The Quadratic Formula
The quadratic formula
gives the solutions x to
where a,b,c are real numbers. It holds even if b2 − 4ac < 0. This is easy to show from the above. There are
exactly two square roots to this number b2 − 4ac from the above methods using De Moivre’s theorem.
These roots are of the form
Thus the solutions, according to the quadratic formula are still given correctly by the above
Do these solutions predicted by the quadratic formula continue to solve the quadratic equation? Yes,
they do. You only need to observe that when you square a square root of a complex number z, you recover
Similar reasoning shows directly that
also solves the quadratic equation.
What if the coefficients of the quadratic equation are actually complex numbers? Does the formula hold
even in this case? The answer is yes. This is a hint on how to do Problem 28 below, a special case of
the fundamental theorem of algebra, and an ingredient in the proof of some versions of this
Example 1.8.1 Find the solutions to x2 − 2ix − 5 = 0.
Formally, from the quadratic formula, these solutions are
Now you can check that these really do solve the equation. In general, this will be the case. See Problem 28