Algebra Know-It-ALL

The fact that the discriminant is positive tells us that this quadratic has two distinct real

roots. To find the roots, we can use the quadratic formula:

x= [−b± (b^2 − 4 ac)1/2] / (2a)

We already know the discriminant, so we can plug it in directly along with the values for

a,b, and c, getting

x= [− 3 ± 289 1/2] / [2 × (−2)]

= (− 3 ± 17) / (−4)

= (− 3 + 17) / (−4) or (− 3 − 17) / (−4)

= 14 / (−4) or (−20) / (−4)

=−7/2 or 5

The real roots are x=−7/2 or x= 5, and the real-number solution set is {−7/2, 5}. For

complementary credit, you can check these roots by plugging them into the original qua-

dratic to be sure that they work.

10. Once again, we can calculate the discriminant to find out how many real roots there
    are. The equation of interest is

4 x^2 +x+ 3 = 0

Here, a= 4, b= 1, and c= 3. Therefore

b^2 − 4 ac= 12 − 4 × 4 × 3

= 1 − 48

=− 47

Because this is negative, we can conclude that the quadratic has no real roots. The real-

number solution set is therefore ∅, the empty set. But this does not mean that there are

no roots at all! In the next chapter, we’ll learn about the roots of quadratics that have

negative discriminants.

Chapter 23

1. We’ve been told to find the roots of this quadratic:

(x−j7)(x+j7)= 0

Because this equation is in binomial factor form, the roots can be found by solving these

two first-degree equations:

x−j 7 = 0

Chapter 23 675