Chapter 9
Complex Numbers and the Quadratic Formula
Square any number you have ever used, positive or negative, and the answer is never below zero. That looks settled until a quadratic equation runs perfectly for several steps and then asks for a number whose square is negative, at which point the work simply stops. Either that equation has no answer at all, or the numbers you have been allowed to use are missing one. Taking that second possibility seriously is what this chapter does. It builds the missing number, learns to compute with it, then rebuilds quadratic-solving itself so every quadratic finally has an answer.
What You'll Explore
5 lessons.
Some lessons are marked Advanced because they go beyond core Algebra I. Skip them or explore them if you are curious.
- Imaginary Numbers Advanced. This lesson goes beyond core Algebra I. You can skip it.
No real number squares to something negative, so the square root of a negative has nothing to point at. Suppose you allow such a number anyway: this lesson follows what that decision costs and what it buys, from a repeating pattern in its powers to one radical rule that stops behaving.
- Arithmetic with Complex Numbers Advanced. This lesson goes beyond core Algebra I. You can skip it.
Once a new number joins the old ones, every operation has to be asked about again. Adding two of them feels almost obvious, multiplying takes more thought, and dividing looks impossible at first, so this lesson works out how each one behaves.
- Completing the Square
Some quadratics factor in seconds while others resist every guess, leaving factoring a method you cannot always count on. This lesson asks whether any quadratic can be reshaped until a perfect square sits inside it, and what the roots look like once that is done, including the ones that are not real numbers.
- The Quadratic Formula
Every quadratic you have solved so far took its own round of work, even though the steps hardly changed. What if that work were done once, in general, so any quadratic could be settled straight from its coefficients? A second question follows: can the kind of answer be known before the solving starts?
- Applications of Quadratics
Throw a ball and it rises, slows, and falls back; fence a garden and its area depends on the shape you pick. Both lead to the same kind of equation, and this lesson takes up how a description in words becomes one. Solving it often hands back two numbers, and the real question is how the situation decides whether one, both, or neither of them is the answer you actually want.