Algebra Know-It-ALL

506 Review Questions and Answers

Answer 23-2

First, we must get the equation into polynomial standard form. We can do that by adding x

to both sides and then switching the right and left sides, getting

3 x^2 +x+ 4 = 0

In the general polynomial standard equation

ax^2 +bx+c= 0

we have a= 3, b= 1, and c= 4. Plugging these into the quadratic formula, we get

x= [−b± (b^2 − 4 ac)1/2] / (2a)

= [− 1 ± (1^2 − 4 × 3 × 4)1/2] / (2 × 3)

= [− 1 ± (1 − 48)1/2] / 6

= [− 1 ± (−47)1/2] / 6

The quantity (−47)1/2 is the imaginary number j(471/2). Therefore

x= [− 1 ±j(471/2)] / 6

If we want to express these roots individually, we can write

x= [− 1 +j(471/2)] / 6 or x= [− 1 −j(471/2)] / 6

We can reduce these to standard complex-number form using the right-hand distributive law

for division over addition or subtraction, getting

x=−1/6+j(471/2/6) or x=−1/6−j(471/2/6)

Question 23-3

Suppose we examine the discriminant of a quadratic equation, and we discover that the equa-

tion has two complex roots. How can we tell whether or not these are pure imaginary num-

bers?

Answer 23-3

If the discriminant is a negative real and the coefficient of x is 0, then the roots are pure imagi-

nary. If the discriminant is a negative real and the coefficient of x is a nonzero real, then the

roots are complex but not pure imaginary.

Question 23-4

In a quadratic equation with real coefficients but pure imaginary roots, how are those roots

related? In a quadratic equation with real coefficients but complex roots that aren’t pure imagi-

nary, how are those roots related?