The converses of these four cases are also true. For instance, if the roots of (1) are imaginary, from (2) and (3) it is clear that 62 4 ac must be negative.

The expression ▲ = b2 — 4 ac is called the discriminant of the equation ax2 + bx + c = 0.

EXERCISES

1. Determine the nature of the roots of the following equations without solving.

.

Solution: A = ( — 4)2 — 4 · 3 ·⋅ ( − 1) = 16 + 12 = 28 and is then positive. Thus by III the roots are real and distinct.

2. Determine real values of k so that the roots of the following equations may be equal. Check the result.

Since the roots of an equation are equal when and only when its discriminant equals zero (§ 98, II), the required values of k make ▲ = 0 and are the roots of

4k2-4k-8 = 0,

CHAPTER IX

GRAPHICAL REPRESENTATION

99. Representation of points on a line. Let us select on the indefinite straight line AB a certain point O as a point of reference. Let us also select a certain line, the length of which for the purpose in hand shall represent unity. Let us further agree that positive numbers shall be represented on AB by points to the right of 0, whose distances from O are measured by the given

3,,

numbers, and negative numbers similarly by points to the left. Then there are certainly on AB points which represent such numbers as 2, - 1, or, in fact, any rational numbers. Since we can divide a line into any desired number of equal parts, we are able to find by geometrical construction the point corresponding to any rational number. Furthermore, by the principle that the square of the hypotenuse of a right triangle equals the sum of the squares of the other two sides, we can find the point corresponding to any irrational number expressed by square-root signs over rational numbers. More complicated irrational numbers cannot, however, in general be constructed by means of ruler and compasses, but we assume that to every real number there corresponds a point on the line, and conversely, we assert that to every point on the line corresponds a real number. This assumption of a one-to-one correspondence between points and real numbers is the basis of the graphical representation of algebraic equations.

This amounts to nothing more than the assertion that every real number, rational or irrational, as, for instance, 6, 2 + √3, √3, π, represents a certain distance from O on AB, and conversely, that whatever point on the line we may select, the distance from 0 to that point may be expressed by a real number..

Y axis

100. Cartesian coördinates. We have seen that when the single letter x takes on real values all these values may be represented by points on a straight line. When, however, we have two variables, as x and y, which we wish to represent simultaneously, we make use of the plane. Just as we determined arbitrarily, on the line along which the single variable was represented, an arbitrary point for the point of reference and an arbitrary length for the unit distance,

1

X axis

so now we select an indefinite line along which x shall be represented, and another perpendicular to it along which y shall be represented. The former we call the X axis; the latter the Y axis. The intersection O of these axes we take as the point of reference for each. This point is called the origin.

We select a unit of distance for x and a unit of distance for y which may or may not be the same, according to the problem under discussion. As before, we represent positive numbers on the X axis to the right, and negative numbers to the left. Positive values of y are represented above the X axis, and negative values below it. The arrowhead on the

(4,3)

(-3.1)

0(0,0) (3.0)

(4,-2)

axes indicates the positive direction. Any pair of values of x and y, written (x, y), may now be represented by a point on the plane which is x units from the Y axis and y units from the X axis. Thus if x = 0, y = 0, written (0, 0), the point represented is the origin. The point (3, 0), i.e. x = 3, y= 0, is found by going three units of x to the right, i.e. in the positive direction of x and no units up. (4, 3), i.e. x = 4, y = 3, is found by going four units of x to the right and three units of y up. The point (-3, 1) is found by

(-3,-4)

The point

going three units of x to the left and one unit of y up. The point (−3, 4) is found by going three units of x to the left and four units of y down. In fact, if we let both x and y take on every possible pair of real values, we have a point of our plane corresponding to each pair of values of (x, y). Conversely, to every point of the plane correspond a pair of values of (x, y). These values are called the coördinates of the point. The value of x, i.e. the distance of the point from the Y axis, is called its abscissa; the value of y, i.e. the distance of the point from the X axis, is called its ordinate. If the point (x, y) is conceived as a moving point, and if no restriction is placed on the value of the coördinates so that they take on every possible pair of real values, every point in the plane is reached by the moving point (x, y).

The X and Y axes divide the plane into four parts called quadrants, which are num

2nd Quadrant 1st Quadrant (+) (++)

Χ

3rd Quadrant 4th Quadrant (+-)

(-6-)

bered as in the figure. The proper signs of the coördinates of points in each of the quadrants are also indicated.

EXERCISES

The following exercises should be carefully worked on plotting paper, which can be bought ruled for the purpose.

– 2), (2,

4).

1. Plot the points (2, 3), (0, 4), (— 4, 0), (— 9, 2. Plot with the aid of compasses the points (1, √2), (√3, − √2),

(2 + √3, 2√3) (− √2, − √2).

− 1, − 1), (− 1,+ 1),

3. Plot the square three of whose vertices are at (+1, − 1). What are the coördinates of the fourth vertex?

4. Plot the triangle whose vertices are (2, 1), (— 6, — 2), (— 4, 4). 5. Plot the two equilateral triangles two of whose vertices are (6, 1), (-6, 1). Find coördinates of the remaining vertices.

6. If the values of the coördinates (x, y) of a moving point are restricted so that both are positive and not equal to zero, where is the point still free to move?

7. If the coördinates (x, y) of a moving point are restricted so that continually y = 0, where is the point still free to move?