CHAPTER XXVI. PROPERTIES OF QUADRATICS. 259. A quadratic equation has two, and only two, roots. Consider the equation x2 + px = q. Suppose this equation to have three different roots, 71, 72 and r; that is, three different values for x. (6) Subtracting (2) from (1), we have, Hence, i r22 — r2 + pr1 − pr2 = 0 = (~1 − 7°2) (~1 + 72) + p (r1 − No2) = 0 = (r1 − r2) (r1 + 1⁄2 + p) = 0. r2 = 0, or r1+ r1⁄2 + p = 0; but as the roots were different, by hypothesis, r1r cannot equal zero. Hence, r1 + r2+ p = 0. In like manner, from (3) and (1), we have, r1+r3+ p = 0. (7) (8) Now, subtracting (7) from (8), we have, r1⁄2 — r = 0, which is impossible. Hence, a quadratic equation cannot have more than two roots. Again, consider the equation x2 + px = q. Let r1 and r1⁄2 be its roots. N 260. Hence, if a quadratic equation is of the form x2 + px = q, the sum of the roots is equal to the coefficient of x with opposite sign, and the product of the roots is equal to the second member with opposite sign. Again, substituting the values of p and q in the equation x2 + px = q, we have, X2 − (~1 + 12) X + r1r2 = = 0. which, we have, Factoring That is, any quadratic equation of the form x2+px = q may be written, (x − r1) (x − r2) = 0. 261. Hence, when the roots of a quadratic equation are given, the equation may be formed by subtracting each root from x, and placing the product of the resulting expressions equal to zero. CHARACTER OF THE ROOTS OF A QUADRATIC EQUATION. The general quadratic equation ax2 + bx + c = 0 gives the -b+b2 4 ac roots, x= It is evident that, I. If b2 — 4 ac is positive and greater than zero, the roots are real and unequal. II. If b2 - 4 ac equals zero, the roots are real and equal. EXAMPLES: (1) Find the sum and the product of the roots of Hence the sum of the roots is equal to 3, and the product is cqual to . EXERCISE 95. Find by inspection the sum and the product of the roots of the following equations: 10. (m+1)2 + (m + 1) x + (m + 1)2 = 0. 11. (a+b)2 + ( a − b) x = − (a2 + b2). (a (2) Find the quadratic equation whose roots are and (3) Determine the nature of the roots of the quadratic Hence, a = 3x2-2x+4. ວາ b = 2, c = 4. b2 - 4 ac = 4 — 48 — —44. Therefore, the roots are imaginary. (6) Find the values for m, for which the following equation has two equal roots: Here, (m − 2) x2 + (m5) x + 2m - 50. a = m 2, b = m − 5, c = 2 m — 5. Now, if the roots are to be equal, b2-4ac (m-5)2 - 4 (m2) (2m 5) = 0 = Substituting these values in the given equation, we have, Determine without solving the character of the root of the following quadratic equations: 1. x2-9x + 20 = 0. 7. x2+8x+16= 0. Determine the values for m in the following quadratic equations which shall give equal values for x: 1. (3m+1)x2 + 2 mx + 2 x + m = 0. |