High School: Number and Quantity
The Complex Number System HSN-CN.C.7
7. Solve quadratic equations with real coefficients that have complex solutions.
When students dealt with parabolas, they only saw the good parabolas. The ones with nice simple roots like -1 and 1. The roots could be found just by looking at the graph or, if they were ugly, by using the quadratic formula.
They probably didn't think too much about those "other" parabolas. The parabolas from the wrong side of the tracks. Yeah, we mean the ones that don't cross the x-axis at all. Your students didn't consider that those parabolas have roots (and feelings), too.
Well, it's time. They're old enough to know the truth.
Those quadratic equations have imaginary roots. Now that your students know that "imaginary" doesn't mean "make believe," we can make it up to these forgotten, mistreated parabolas. Flowers and chocolate just won't cut it; we'll have to work with them.
Let's take a look at one: x2 + 4x + 6 = 0. This one doesn't factor, so we'll have to use the quadratic formula:
For this particular parabola, our values are a = 1, b = 4 and c = 6. When we plug in those numbers, we get
But wait. We can't have the square root of a negative number if we're working with real numbers. Good thing we're working with imaginary numbers, then, isn't it?
That means we can turn that fraction into which simplifies to . So our parabola has 2 perfectly lovely roots. The only problem is that the graph doesn't cross the x-axis because it only contains real numbers, and the roots are imaginary, not real.
Quadratics with imaginary roots aren't always easy to identify until we get to the point in the problem where we have to take the square root. At that point, if the number under the radical is negative, that parabola's roots will be imaginary. (That number under the radical, b2 – 4ac, is called the discriminant. Probably because it discriminates between real and imaginary roots.)
Now, your students should be able to calculate the roots of these other parabolas and tell whether or not a parabola will have real or imaginary roots by looking at its discriminant.