increases without limit, that is, approaches infinity. Thus the quadratic equation approaches a linear equation when a approaches O, and one of its roots disappears since it has increased in value beyond any finite limit. The loop-shaped graph of the quadratic equation must then approach a straight line as a limit when a approaches 0. This is made clear from the following figure, where a has the successive values 1, }, ro, s'o, 0. In the figure the curves represent the following equations: x2 2 = y. 2 = (II) y. (III) 2=y. (IV) In a similar manner we can show that when in the equation bx + c = 0, b approaches 0 as a limit, the root of the linear equation becomes infinite. EXERCISES 1. What real values must k approach as a limit in order that one root of each of the following equations may become infinite? 2. What real values must k and m approach as a limit in order that both roots of the following may become infinite? (f) (k + m) x2 + 2 (k + m) + 1 = x2 − 2 x. (g) (k+m+1) x2 + (2 k − m − 1)x + 1 = 0. (h) (2k + m + 2) x2 + (4 k + 2 m + 3)x + 3 = 0. 115. Sum and difference of roots. Let x and x, be the roots of x2 + bx + c = : 0. x1 and x (1) Then (§ 95) x — x2 are factors, and their product x2 - (x1+x2)x + x1x2 is exactly the left-hand member of (1). Consequently the equation x2 + bx + c = x2 − (x1 + x2) X + X1X2 is true for all values of x. Hence by § 96 We may state these facts in the THEOREM. The coefficient of x in the equation x2 + bx + c = 0 * is equal to the sum of its roots with their signs changed. The constant term is equal to the product of the roots. EXERCISES 1. Prove the statement just made from the expression for the roots in terms of the coefficients (§ 89). 2. Form the equations whose roots are the following: and * We should for the present exclude the case where b2 - 4c <0, since the roots x1 x, are then imaginary and we have not as yet defined what we mean by the sum or the product of imaginary numbers. We shall see later that the theorem is also true in this case. 3. If 4 is one root of x2 − 3x + c = 0, what value must c have? 4. Find the value of the literal coefficients in the following equations. 116. Variation in sign of a quadratic. It is often necessary to know the sign of the expression ax2 + bx + c for certain real values of x, and to determine the limits between which x may vary while the expression preserves the same sign. We assume as usual that a is positive. THEOREM I.* If the discriminant of ax2 + bx + c is positive, the quadratic is negative for all values of x between the values of the roots of the equation. For other values of x (excepting the roots) the quadratic is positive. * If a were negative, Theorem I would read as follows: If the discriminant is positive, the quadratic is positive for all values of x between the values of the roots of the equation. For other values of x (excepting the roots) the quadratic is negative. When a is negative Theorems II and III may be modified in an analogous manner. In § 98 we found that when the discriminant of a quadratic equation is positive the equation has two real roots. If two roots are real, the loop of the graph of the equation ax2 + bx + c = y cuts the X axis in two points Y A P RO (§ 110) as in the figure. The roots are represented by A and B, and any real value of x between the roots is represented by a point P in the line AB. Since the curve is below the X axis at any such point, the value of y, i.e. of the expression ax2 + bx + c for values Xof x between the roots, is negative. The value of the expression for any value of x greater or less than both roots is seen to be positive, since for such points, for example Q and R, the graph is above the X axis. THEOREM II. If the discriminant of ax2 + bx + c is negative, the expression is positive for all real values of x. When the discriminant is negative the entire graph of ax2 + bx + c = y is above the X axis (§ 111), and consequently for any real value of x the corresponding value of y, i.e. the value of ax2 + bx + c, is positive. THEOREM III. If the discriminant of ax2 + bx + c is zero, the value of the expression is positive for all values of x except the roots of the equation ax2 + bx + c = 0. Hint. See example 16, p. 102. We may restate these three theorems and prove them algebraically as follows: THEOREM IV. If the discriminant of the quadratic ax2 + bx + c is positive, the values of the quadratic and a differ in sign for all values of x lying between the roots, and agree for other values. If the discriminant is zero or negative, the value of the quadratic always agrees with a in sign. CASE I. Since the discriminant is positive, the equation ax2+bx+c=0 has two unequal real roots, as x1 and x2, of which we will assume x1 is the greater, and we may write the quadratic in the form ax2 + bx + c = α (x − x1) (x — X2). Now for any value of x between x1 and x2 the factor x x1 is negative, while x - x2 is positive, which shows that the quadratic is opposite in sign to a for such values of x. For other values of x both these factors are either positive or negative, and for such values the quadratic is of the same sign as a. member of the equation has the same sign as a. CASE III. Since the discriminant is zero, the roots are equal and the expression has the form ax2 + bx + c = a (x − x1)2, which has evidently the same sign as a, for any value of x. EXERCISES 1. Between what values of x is the expression √x2 Solution: The roots of x2 5x40 are 4 and 1. The discriminant ▲ = 12 4 ac = 25 169 is positive. Thus by Theorem I or IV, if 1<<4† the expression under the radical sign is negative and the whole expression is imaginary. 2. For what values of k are the roots of (k + 3) x2 + kx + 1 = 0 (1) (a) real and unequal? (b) imaginary ? Solution: a = k+ 3, bk, c = 1. Ab2-4ac = · k2 — 4 (k + 3) = k2 - 4k - 12. (a) If ▲ > 0, the roots of (1) are real and unequal. The roots of k2 4k 12 are k = 6 and 2. * See § 152. † Read"1 is less than x which is less than 4" or "x is between 1 and 4." |