If 1, w, w2 are the three cube roots of unity, prove 24. (1+w2)4=w. ய. 25. (1-w+w2) (1+w− w2) = 4. 26. (1) (1- w2) (1 - w1) (1 - w3)=9. 27. (2+5w+2w2)6=(2+2w+5w2)6729. 28. (1−w+w2)(1 − w2 + w1) (1 − w1+w3)..... to 2n factors = 22n. 29. Prove that 23+y3+23-3xyz=(x+y+z) (x+yw+zw2) (x+yw2+zw). 30. If shew that (1) xyz=a3+63. (2) 22+y2+22=6ab. (3) x3+y3+z3 = 3 (a3+b3). 31. If ax+cy+bz=X, cx+by+az=Y, bx+ay + cz=Z, shew that (a2 + b2 + c2 — bc - ca - ab) (x2+y2+z2 — yz — zx — xy) CHAPTER IX. THE THEORY OF QUADRATIC EQUATIONS. 111. AFTER suitable reduction every quadratic equation may be written in the form We shall now prove some important propositions connected with the roots and coefficients of all equations of which (1) is the type. 112. A quadratic equation cannot have more than two roots. For, if possible, let the equation ax2 + bx + c = 0 have three different roots a, ß, y. Then since each of these values must satisfy the equation, we have divide out by a-ẞ which, by hypothesis, is not zero; then a (a + B)+b=0. Similarly from (2) and (3) a (ẞ + y) + b = 0; (1), (2), (3). .. by subtraction not equal to y. a (a− y) = 0; which is impossible, since, by hypothesis, a is not zero, and a is Hence there cannot be three different roots. 113. B, so that In Art. 111 let the two roots in (2) be denoted by a and then we have the following results : (1) If b2-4ac (the quantity under the radical) is positive, a and ẞ are real and unequal. (2) If b2-4ac is zero, a and ẞ are real and equal, each reducing in this case to b 2a (3) Ifb2-4ac is negative, a and B are imaginary and unequal. (4) If b2 - 4ac is a perfect square, a and ẞ are rational and unequal. By applying these tests the nature of the roots of any quadratic may be determined without solving the equation. Example 1. Shew that the equation 2x2-6x+7=0 cannot be satisfied by any real values of x. Example 2. If the equation x2 + 2 (k + 2) x + 9k=0 has equal roots, find k. The condition for equal roots gives (k+ 2)2=9k, k2 - 5k+4=0, (k − 4) (k − 1) = 0; .. k=4, or 1. Example 3. Shew that the roots of the equation are rational. x2 - 2px +p2 − q2+2gr — r2 = 0 The roots will be rational provided (--2p)2 - 4 (p2 - q2+2gr - r2) is a perfect square. But this expression reduces to 4 (q2 - 2gr +r2), or 4 (g−1)2. Hence the roots are rational. In a quadratic equation where the coefficient of the first term is unity, (i) the sum of the roots is equal to the coefficient of x with its sign changed; (ii) the product of the roots is equal to the third term. NOTE. In any equation the term which does not contain the unknown quantity is frequently called the absolute term. or с a O may be written Hence any quadratic may also be expressed in the form x2 - (sum of roots) x + product of roots = 0.........(2). Again, from (1) we have (x − a) (x − ẞ) = 0 We may now easily form an equation with given roots. Example 1. Form the equation whose roots are 3 and -2. (x − 3) (x+2)=0, x2-x-6=0. (3). When the roots are irrational it is easier to use the following method. Example 2. Form the equation whose roots are 2+√3 and 2√3. 116. By a method analogous to that used in Example 1 of the last article we can form an equation with three or more given roots. Example 1. Form the equation whose roots are 2, – 3, and 7 The required equation must be satisfied by each of the following suppositions: с a, 1. Example 2. Form the equation whose roots are 0, ±a, 117. The results of Art. 114 are most important, and they are generally sufficient to solve problems connected with the roots of quadratics. In such questions the roots should never be considered singly, but use should be made of the relations obtained by writing down the sum of the roots, and their product, in terms of the coefficients of the equation. Example 1. If a and ẞ are the roots of x2 − px+q=0, find the value of (1) a2 +ẞ2, (2) a3 +ß3. |