Algebra Know-It-ALL

508 Review Questions and Answers

Because p and q are both positive, we know that the ratio q/p is positive. Its positive and nega-

tive square roots are therefore both real numbers. We can take the square root of both sides of

the above equation, getting

x=±[−1(q/p)]1/2

=±(−1)1/2 [±(q/p)1/2]

=±j[(q/p)1/2]

The roots are therefore x=j[(q/p)1/2] or x=−j[(q/p)1/2].

Question 23-7

Is it possible for the roots of a quadratic equation to be pure imaginary but not additive

inverses? If so, provide an example of such an equation. If not, explain why not.

Answer 23-7

A quadratic equation can have roots that are pure imaginary but not additive inverses. Here is

an example of such an equation in binomial factor form:

(x+j2)(x+j3)= 0

The roots of this equation are x=−j2 or x=−j3, as we can verify by plugging them in. They

are not additive inverses.

Question 23-8

We have learned in the last several chapters (but not explicitly stated in full, until now!), that if

the polynomial standard form of a quadratic equation has real coefficients and a real constant,

then one of these things must be true:

• There are two different real roots


• There is a single real root with multiplicity 2


• There are two different pure imaginary roots, and they are additive inverses


• There are two different complex roots, and they are conjugates

In Answer 23-7, we found a quadratic equation that has two pure imaginary roots that are not

additive inverses. How is this possible?

Answer 23-8

The coefficients and constant in the polynomial standard form of this equation are not all real

numbers. To see that, we can multiply the product of binomials out:

(x+j2)(x+j3)=x^2 +j 3 x+j 2 x+ (j 2 j3)

=x^2 +j 5 x− 6