where as usual a, b, and c represent integers and a is positive. If we let x take on various values, y will have corresponding values and we may plot the equation as in § 103. A root of the quadratic equation ax2 + bx + c = 0 (2) is a number which substituted for x satisfies the equation, that is, gives the value y = 0 in (1). Thus the points on the graph of (1) which represent the roots of the equation (2) are the points for which y = 0, that is, where the curve crosses the X axis. The numerical value of the roots is the measure of the distance along the X axis from the origin to the points where the curve cuts the axis. Since this distance is always a real distance, only real roots are represented in this manner. THEOREM. If the graph of (1) has no point in common with the X axis, the equation (2) has imaginary roots, and conversely. Every equation of form (2) has two roots either real or imaginary (§ 89). If the graph of (1) has no point in common with the X axis, there is no real value of x for which y = 0, i.e. no real root of (2). The roots must then be imaginary. Conversely, if (2) has only imaginary roots, there is no real value of x which satisfies it, i.e. which makes y = 0 in (1). Thus the curve has no point in common with the X axis. This suggests the following universal PRINCIPLE. Non-intersection of graphs corresponds to imaginary or infinite-valued solutions of equations. 111. Form of the graph of a quadratic equation. Consider the equation y = 2x2 + 7 x + 2. (1) By substituting for x a very large positive or negative number, say x = 100, y is large positively. Thus for values of x far to the right or left the curve lies far above the X axis. If we assign to y a certain value, say y = 2, we can find the corresponding values of x by solving a quadratic equation. Thus in (1) let y = or = 2. The roots are 2 = 2x2 + 7 x + 2, 2x2+7x=0. - x Hence the points (— 31, 2) and (0, 2) are on the curve (§ 101). That is, if we go up two units on the Y axis, the curve is to be found three and one half units to the left and also again on the X axis. If in (1) we let y = 4, the corresponding values of a are very nearly equal to each other (-1 and 2), which means that the curve meets a line parallel to the X axis and four units below it at points very near together. The question now arises, Where is the bottom of the loop of the curve? This lowest point of the loop has as its value of y that number to which correspond equal values of x. Hence we must determine for what value of the equation (1), that is, the equation y has equal roots. (§ 98), we have or 2x2+7x+(2 − y) = 0, Comparing with the equation ax2 + bx + c = 0 Thus the condition 2 - 4 ac 0 becomes b, 2 - y = c. Substituting this value of y in (1), we get that the bottom of the loop is at a 4 ac 4 a A 4 a Thus we see again that if the discriminant is negative the graph is entirely above the X axis and both roots are imaginary (§§ 98, 110), since the ordinate of the lowest point of the loop is positive. If the discriminant is positive, the graph cuts the X axis and both roots are real. The results of this section enable us to determine a value of y from the coefficients which determine the lowest point of the loop of the curve precisely, and hence to show beyond question from the graph whether the equation has real or imaginary roots. EXERCISES Plot the following equations and determine by measurement the roots in case they are real. Find in each case the lowest point on the loop. 16. What is the characteristic feature of the plot of an equation whose roots are equal? 112. The special quadratic ax2 + bx = 0. When in the quad c = 0, we can always factor the equation into (1) - Conversely, if x = 0 is a root, then (§ 95) x 0, or x, is a factor and the equation can have no constant term. This affords the THEOREM. A quadratic equation has a root equal to zero when and only when the constant term vanishes. We show in a similar manner that both roots of the equation (1) are zero when and only when b = c = 0. EXERCISES 1. Prove the theorem just given by considering the expressions for the roots in terms of the coefficients (§ 89). 2. For what real values of k do the following equations have one root equal to zero ? (a) x2+6x- k + 1 = 0. (c) x2 + 6x + k2 + 1 = 0. (b) 2x2 - 3x + k2 − 1 = 0. (d) 2 x2 4x+k2-3 k = 0. (e) 2x2+2kx - 2 k2 - 4 k − 2 = 0. (f) 6 x2 - 4 x + 2 k2 + k + 7 = 0. 3. What is the characteristic feature of the plot of an equation which has one root equal to zero? 4. For what real value of k will both roots of the following equations vanish? 113. The special quadratic ax2 + c = 0. This equation may с be written in the form x2 + = 0 and factored immediately into a which shows that the roots are equal numerically but have oppo Since in the equation ax2 + c = y the variable x occurs only in the term x2, we get the same value of y for positive and negative values of x. Hence the loop which forms the graph of the equation is symmetrical with respect to the Y axis. 114. Degeneration of the quadratic equation. The equation ax2 + bx + c = 0 We wish to find the effect on the roots x and x, when a becomes very small. If we let a approach 0, then x1 approaches 0 0' an expression of the form which must always be avoided. Rationalize the numerators and we get As a approaches 0, evidently b2 - 4 ac approaches b2, x1 ap proaches, and x2, since its denominator becomes very small, с a is positive this involves real factors. If - is negative the factors are * When imaginary (§ 152). a |