Algebra Know-It-ALL

384 Quadratic Equations with Complex Roots

which simplifies to

ax^2 +c= 0

Subtracting c from each side gives us

ax^2 =−c

We can divide through by a because, in a quadratic, a is never 0. Doing that, we get

x^2 =−c/a

Taking the positive-or-negative square root of each side, we obtain

x=±[(−c/a)1/2]

If a and c have opposite signs, then −c / a is positive. Therefore, the roots are both real and are

additive inverses of each other:

x= (−c/a)1/2 or x=−[(−c/a)1/2]

If a and c have the same sign, then −c /a is negative. That means the roots are pure imaginary

and are additive inverses of each other:

x=j(|−c/a|1/2) or x=−j(|−c/a|1/2)

In this case, the discriminant is

d=b^2 − 4 ac

= 02 − 4 ac

=− 4 ac

Because a and c have the same sign, − 4 ac< 0. Therefore, d< 0.

Conjugate roots

The discriminant in a quadratic can be negative even when b, the coefficient of x, is not equal

to 0. The only requirement is that 4ac be larger than b^2. Here’s an example:

(45/2)x^2 + 3 x+ 1 = 0

In this case, a= 45/2, b= 3, and c= 1, so we have

4 ac= 4 × (45/2) × 1

= 90