Algebra Know-It-ALL

530 Review Questions and Answers

The discriminant of this quadratic is negative, telling us that it has no real roots. The

x-values of any solutions we can derive for the original system will not be real numbers. There-

fore, no real solutions exist.

Chapter 28

Question 28-1

How do we graph a general two-by-two system of equations when we want to see approxi-

mately where the curves intersect at the real solutions, but we don’t need a lot of precision?

Answer 28-1

First, we can calculate several ordered pairs for both functions individually, including the real

solutions, if any exist. It can be helpful to put the values in a table. Next, we figure out the

scales we should have on our graph so as to provide a good “picture” of the situation. Then we

plot the real solution point or points, if any exist, on the coordinate grid. After that, we plot

the rest of the points based on the values in the table we’ve created. Finally, we fill in the lines

or curves for both functions.

Question 28-2

Consider the system of equations we solved in Answer 27-2:

y=x^2

and

y=−x^2

How can we sketch an approximate graph of this system, showing the real solution?

Answer 28-2

We can tabulate and plot several points in both functions including the real solution, (0,0).

Table 30-1 compares some values of x, some values of the first function, and some values of

Table 30-1. Selected values for graphing the functions

y=x^2 and y=−x^2.

The bold entry indicates the real solution.

xx^2 −x^2

−2 4 − 4

−1 1 − 1

0 0 0

1 1 − 1

2 4 − 4