and
b^2 = 32
= 9
Therefore,
d=b^2 − 4 ac
= 9 − 90
=− 81
This tells us that the roots of the quadratic are not real numbers. To find the roots, we can plug
in the value −81 for d in the “abbreviated discriminant” form of the quadratic formula:
x= [−b±j(|d|1/2)] / (2a)
= [− 3 ±j(|−81|)1/2] / [2 × (45/2)]
= [− 3 ±j(811/2)] / 45
= (− 3 ±j9) / 45
=−3/45±j(9/45)
=−1/15±j(1/5)
The roots are
x=−1/15+j(1/5) or x=−1/15−j(1/5)These are complex conjugates. As things work out, the roots are always complex conju-
gates in a quadratic where d< 0, even when b≠ 0. The solution set X in this example is
X= {−1/15+j(1/5),−1/15−j(1/5)}Are you ambitious?
For complementary credit, plug the roots we’ve just found into the original quadratic to be sure that they
work. You’re on your own. Here’s a hint: This is a messy process, but if you’re careful and patient, all the
“garbage” will drop out in the end.
Are you confused?
The above derivations are abstract, but it’s important that you follow through them so you understand the
reasoning behind each step. Here are the results, wrapped up into two statements. In any quadratic:
- If the discriminant is a negative real number and the coefficient of x is 0, then the roots are pure
imaginary, and are additive inverses. - If the discriminant is a negative real number and the coefficient of x is a nonzero real number, then
the roots are not pure imaginary, but are complex conjugates.
Complex Roots by Formula 385