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