Relations Between Roots and Coefficients 5. Representing the two roots of the standard quadratic equation ax2 + bx + c = 0 by as in § 1, above, it follows immediately by addition that 2 a (1) b-b2 4 ac 2 a If the terms of a quadratic equation in standard form, ax2 + bx + c = = 0, (2) be all divided by a, which is the coefficient of x2, the coefficient of will be unity in the derived equivalent equation Referring to (1) and (2), it may be seen that in (3): (i.) The sum of the roots x and x is equal to the coefficient of x with its sign changed b a (ii.) The product of the roots is equal to the term free from x, Principles (i.) and (ii.) may be used as checks upon the solution of a quadratic equation. E. g. The roots of 2 x2 + 7x+3=0 are -3 and 1/2. Dividing the terms by 2, which is the coefficient of x2, we obtain the It may be seen that the sum of the roots 3 and of x with its sign changed, that is, -, and the product of the roots is . с 6. From 12 = (2), obtained in § 5, it appears, if the roots 1 and 2 are real, that according as the numbers represented in the standard equation by a and c have like or unlike signs, the quotient will be positive or negative, and the roots of the given с a equation will be both positive or both negative. E. g. Consider 2 x2 7x+3=0. Using the discriminant we find that the roots cannot be imaginary, since (-7)2 - 4 · 2 · 325, which is a positive number. . a The quotient, represented by, is positive. Hence the roots cannot differ in sign, and must be either both positive or both negative. It may also be seen that the roots of the equation 2x2 + 5 x − 3 = 0 are also both real, but they have opposite signs because 7. From x1 + x 2 = roots is represented by sign with the quotient 3 is negative. 2 b with its sign changed, or if they differ in a Hence the roots either both agree in sign, the sign of the greater root must agree with the reversed sign of From this it follows that, if the number represented by a in the standard quadratic equation ax2 + bx + c = 0 is positive, the root which is numerically the greater is opposite in sign to the sign of the number represented by b. E. g. The roots of 2 x2-5x-3=0 are both real, but they have oppo The greater root must be positive because its sign must agree with the reversed sign of the quotient - §. For convenience of reference, the illustrations used above are given in tabular form as follows: Formation of an Equation having Specified Roots 8. A quadratic equation having specified roots may be constructed by applying Principles (i.) and (ii.) §5. b We may substitute x1 + x2 for and a for in the quadratic a equation+x+ = 0 and obtain a a a A quadratic equation having specified roots may be obtained by constructing a quadratic equation of which the coefficient of a is unity, the coefficient of x is the sum of the roots with sign changed, and the term free from a is the product of the specified roots. Whenever fractions appear in any of the terms of an equation thus constructed an equivalent integral equation may be obtained by multiplying the terms by the lowest common multiple of all the denominators of the fractions. Ex. 1. Construct the quadratic equation the roots of which are 5 and 7. The sum of the roots is 5 +7 = 12, and the product is 5 7 = 35. Hence, changing the sign of the sum 12, we may write as the required equation, x2 12 x + 35 = 0. Ex. 2. Construct the equation the roots of which are + 3 and — 8. Ex. 3. Construct the equation the roots of which are and – 3. Changing the sign of the sum, we may construct the equation Ex. 4. Construct the equation the roots of which are Using the sum, with its sign changed, as the coefficient of x, we have, Ex. 5. Construct the equation the roots of which are the conjugate irrational numbers 2 + √5 and 2 – √5. The sum of these values is (2 + √√5) + (2 − √√/5) = 4. The product is (2 + √/5)(2 − √/5) = 4 − 5 = − 1. Reversing the sign of the sum, we may write the equation x2 - 4x 1 = 0. Ex. 6. Construct the equation the roots of which are the complex Using the sum with its sign changed, we have for the equation _ ! 1 3 9. The roots of a quadratic equation in the standard form ax2 + bx + c = 0 are the roots of the two linear equations formed by equating to zero the two linear factors the product of which is the first member of the equation. (See Chap. XII. § 48.) It follows that, by reversing the process, we may construct a quadratic equation having specified roots by equating to zero the product of the two linear factors which, when equated to zero and considered as equations, have as roots the given values. Ex 7. Construct the equation the roots of which are 5 and 7. We may indicate that these are roots x and x2 of an equation by writing 15 and X2: = 7. 27 0. These two equations taken together are equivalent to the single quadratic (x15)(x-7)=0. equation Performing the indicated multiplication and neglecting subscripts, we obtain 22 12x + 35 = 0. (Compare with Ex. 1, § 8.) 10. It will be found that, whenever the roots are irrational or imaginary, the method of § 8 is to be preferred. |