and further to
x^2 + (b/a)x+c/a= 0
Multiplying through by a, we obtain
ax^2 +bx+c= 0
which is the original general quadratic in polynomial standard form.
Practice Exercises
This is an open-book quiz. You may (and should) refer to the text as you solve these problems.
Don’t hurry! You’ll find worked-out answers in App. C. The solutions in the appendix may
not represent the only way a problem can be figured out. If you think you can solve a particu-
lar problem in a quicker or better way than you see there, by all means try it! Note: Some of
these problems take you a little beyond the material covered directly in the text of this chapter.
Nevertheless, you should be able to solve them using the rules and techniques you’ve been
taught so far. Be patient, and be careful with the signs and j operators!
- What are the roots of the following quadratic equation? What is the solution set X?
(x−j7)(x+j7)= 0
- Convert the equation given in Prob. 1 into polynomial standard form.
- Use the quadratic formula to find the roots of the polynomial equation as it is expressed
in the solution to Prob. 2.
- We haven’t dealt with quadratic equations in which the roots are pure imaginary but
are not additive inverses. However, we can “manufacture” such an equation. Suppose
we want the roots to be
x=j7 or x=−j 3
Write down the binomial factor form of a quadratic with these two roots.
- Convert the equation from the solution to Prob. 4 into polynomial standard form.
- Use the quadratic formula to find the roots of the polynomial equation as it is expressed
in the solution to Prob. 5.
- What are the roots of the following quadratic equation? What is the solution set X?
(x+ 2 +j3)(x− 2 −j3)= 0
- Convert the equation given in Prob. 7 into polynomial standard form.
- Plug the roots from the solution to Prob. 7 into the polynomial standard equation from
the solution to Prob. 8, showing that those roots actually work.
- Convert the following quadratic into polynomial standard form:
(j 2 x+ 2 +j3)(−j 5 x+ 4 −j5)= 0
Practice Exercises 395
