Algebra Know-It-ALL

Part Three 507

Answer 23-4

If the roots are pure imaginary, then they are additive inverses. That is, one is the negative of
the other. If the roots are complex but not pure imaginary, then they are conjugates.

Question 23-5

Consider the following quadratic equation in polynomial standard form:

4 x^2 + 64 = 0

How can this equation be rewritten in binomial factor form?

Answer 23-5

If we subtract 64 from both sides of the equation and then divide through by 4, we get

x^2 =− 16

This tells us that the roots are x=j4 or x=−j4. We can write the negatives of these roots as
constants in a pair of binomials and then set their product equal to 0, obtaining

(x−j4)(x+j4)= 0

That’s the binomial factor form of the original equation.

Question 23-6

Consider the general quadratic equation

px^2 +q= 0

where p and q are both positive real numbers. What are the roots of this equation?

Answer 23-6

Subtracting q from each side, we get

px^2 =−q

Dividing through by p, which we know is not 0 because we’ve been told that it’s positive, we
obtain

x^2 =−q/p

which can be rewritten as

x^2 =−1(q/p)