CHAPTER VI RELATIONS BETWEEN THE COEFFICIENTS AND THE ROOTS OF AN EQUATION OF THE SECOND DEGREE 421. It has already been learned that the equation of the second degree has two roots, i. e., that there are two expressions involving the coefficients, a, b, c, of the equation which will satisfy the equation. It is next to be proved that the equation ax2 + bx + c = 0 has two roots only. Let, if possible, the equation (1) 19 ax2 + bx + c = 0 all different. Since r r 29 37 are roots of have three roots r. r Subtract the second and third equations from the first; then, on dividing the first equation by (",) and the second by (r, − r), which is admissible since r1 and r. r, are both different from zero, 1 On taking the difference between the last pair of equations, it is found that But by hypothesis a is 2 a(r,— r1) = 0. Therefore, an equation of the second degree can have but two roots. 422. By calling x, and x, the roots of the equation x2 x2 + px+9=0, it was shown that the trinomial, x2 + px + q, can be decomposed into two factors of the first degree, If the multiplication of xx, by xx, is performed, the identity x- X2 will result. x2 + px+q=x2 − (x ̧ + x ̧) x + x ̧X2 Since the members of this identity are the same, the coefficients of x in the two trinomials must be equal and, likewise, the constant terms; hence, This result can be established also directly from the formulae for the values of x1 and thus Such are the fundamental relations which connect the roots of an equation of the first degree with its coefficients. They may be stated as follows: If the equation of the second degree is reduced to the form x2 + px + q = 0, (1) the sum of the roots is equal to the coefficient of x with the sign changed, and (2) their product is equal to the constant term. If x1 and x2 x, are the roots of this equation, then according to the preceding rule 423. The practical results of the preceding article are shown as follows: Suppose that the sum of two quantities, x, and x27 is a, and their product is b; then x1 and X2 are the roots of the equation For example, if the sum of x1 and X2 is 7 and their product is 12, the two quantities sought are roots of the equation Similarly, two quantities can be found if it is known that their difference is a and their product b. For, let x1 and -x be the two quantities; then and on applying the preceding rule, x, and x, will be roots of the equation, x2 ax b = 0. For example, if the difference is 3 and the product 28, it is necessary to solve the equation, x2 3 x 28 whose roots are x1=7, x1⁄2 = 4; the two numbers sought are 7 and 4. 424. Observations on the Properties of the Roots of the Quadratic Equation. 1. If the third term of the equation of the second degree x2 + px + q = 0 is negative, the roots are always real, unequal, and opposite in sign; because in this case the quantity 1 p2-4 q is real and, therefore, both roots are real and unequal (2412,1). Since the product of the two roots is equal to the third term q, they must be opposite in sign. Thus, for example, it follows that the equation has two real and unequal roots with opposite signs; for here V p2 - 4q = √/25+ 56, and the product of the roots is 14. Since the sum of the roots is 5, the greatest root in absolute value is negative. These two roots must, therefore, be 7 and +2. 2. In case the quantity q is positive, the sign of p2-4 9 must be determined before it can be decided whether the roots are real or not. If this quantity is positive, then the roots are real and have the same sign, since p2 4 q is less than p2. Since the sum of the roots is - p, then the sign of each root is the opposite of that of p. (a) Consider, for example, the equation 4 q = 49 48 x2 − 7 x + 12 = 0. Since p 1 (b) Consider an example in which the second term also is plus: x2 + 9 x + 20 = 0. The quantity p2 — 4 q: are real. = 81 80+1 is positive, and the roots Since the product of the roots is +20, the roots have the Their sum being 9, they are both negative. roots are 4 and 5. same sign. The 3. In case the constant term q is equal to zero, the quadratic equation has the form. since either factor, placed equal to zero, annuls the product. EXERCISE LXXIII Find by inspection the sum and the product of the roots of the following equations. and their product 17 2 3. 2-3x+2=0. 5. 8x+4x. 7. 2(x-1)= 3 (x + 2) (x-3). 9. 15 x2 are the roots of the equation x2 + px + q = 0, Then, according to the rule, (x − 3) (x+7)=0. Multiplying by 5, That is, (-3)(5+1)=0. 5x28x-21= 0. 23. Without solving the equation, find the sum of the squares and the difference of the squares of the roots of the equation 3x-6x-1= 0. 24. Show that the roots of the equation ax2- (a—c)x−(a+b)=0 are always real if a and b are negative. 25. Find the condition under which x2 + px + q = 0 is double the other. x2 one of the roots of 26. Show that (a+b+c) x2 − 2 (a + b) x + (a + b − c) = 0 has rational roots. 27. Find the rational relation which must connect a, b, c, a', b', c', in order that the two equations ax2 + bx + c = 0 a2x2+ b'x + c' = 0, have a common root. |