Algebra Know-It-ALL

382 Quadratic Equations with Complex Roots

Let’s take an example. Suppose that we work out the discriminant d for a quadratic, and

we find that d=−16. Then the positive square root of d is equal to j4, and the negative square

root of d is equal to −j4.

Here’s another example. Let’s revisit the quadratic stated in Practice Exercise 10 at the end

of Chap. 22:

4 x^2 +x+ 3 = 0

In this case, a= 4, b= 1, and c= 3, so

d=b^2 − 4 ac

= 12 − 4 × 4 × 3

= 1 − 48

=− 47

The positive square root of −47 is j(471/2), and the negative square root is −j(471/2).

Now let’s look at the general case. If d< 0, then |d|> 0. (As you know, it also happens to

be true that if d< 0, then |d|=−d, a fact that we’ll use later in this chapter.) We can express

the positive square root of d as

d1/2=j(|d|1/2)

and we can express the negative square root of d as

−(d1/2)=−j(|d|1/2)

Stated as a “plus-or-minus” expression, we have

±(d1/2)=± j(|d|1/2)

Substituting ±j(|d|1/2) in place of ±(b^2 − 4 ac)1/2 in the quadratic formula, we get

x= [−b±j(|d|1/2)] / (2a)

This equation can be used if and only if the real-number discriminant, d, is negative. It’s

important to remember what “if and only if ” means in this context! We can always use this

formula when d< 0. But we must never use it when d= 0 or when d> 0, because in those

cases, the j operator does not belong there.

Imaginary roots: a specific case

Consider the following quadratic equation in which the coefficient of x^2 is positive, the coef-

ficient of x is equal to 0, and the stand-alone constant is positive:

3 x^2 + 75 = 0