Algebra Know-It-ALL

Part Three 505

Question 22-10

How can we use the quadratic formula to find the roots of the following equation?

−x= 3 x^2 − 4

Answer 22-10

First, let’s 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. That gives us

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 ± 49 1/2) / 6

= (− 1 ± 7) / 6

= 6/6 or −8/6

= 1 or −4/3

The roots of the quadratic equation are x= 1 or x=−4/3.

Chapter 23

Question 23-1

Does a quadratic equation with real coefficients and a real constant, but with a negative dis-
criminant, have any roots at all? If so, what are they like?

Answer 23-1

When a quadratic equation with real coefficients and a real constant has a negative discrimi-
nant, the equation has two roots, both of which are non-real complex numbers.

Question 23-2

How can we use the quadratic formula to find the roots of the following equation?

−x= 3 x^2 + 4