? 51. √(x2 + ax + b2) + √(x2 + bx + a2) = a + b. 55. √(x2 + 2x − 1) + √(x2 + x + 1) = √2 + √3. ? 56. √(x2+ax − 1) + √(x2 + bx − 1) = √a + √b. 58. (x2+1)(x + 2) = 2. 60. = 1 х 4x 64. 2x+1+x√(x2 + 2) + (x + 1) √(x2 + 2x + 3) = 0. 65. x2+3=2√(x3- 2x+2)+ 2x. 66. x +5 +4 = 5 ( * +5x+28). 3 70. x2+3-√(2x2 - 3x+2)=√(x+1). 2. 71. x(x+1)+3√(2x2 + 6x + 5) = 25 — 2x. 2a a 72. xo − 2 √(3xo − 2ax+4)+ 4 = 2 (x + 2 + 1 ). 3 81. √x+ √(x+7) + 2 √(x2 + 7x) = 35 – 2x. 82. x-8(x+1)√x+18x + 1 = 0. 83. 2(x2+ax)*+√x + √(a + x) = b − 2x. 84. x2+2x3- 11x2 + 4x + 4 = 0. x2 + 1 1 (x+1)* − 2* = 91. x2+1= 0. 93. (x-2) (x-3) (x-4)= 1.2.3. 92. nữ +x+n+1=0. 94. (x − 1) (x − 2) (x − 3) – (6 − 1) (6 − 2) (6 − 3) = 0. = 0. 8x3+16x9. 101. x(x-2) x (x2 − 2) = m (x2 + 2mx + 2). 102. 100. x2. 2 3x = 14. (x2 − a2) (x + a) b + (a3 − b2) (a + b) x + (b2 − x3) (b + x) α = 103, a+px + (p-1+ 1) +1 = 0. p 104. (p-1)2 + pœ2 + (p-1+)+1=0. XXII. THEORY OF QUADRATIC EQUATIONS AND QUADRATIC EXPRESSIONS. 334. A quadratic equation cannot have more than two roots. For any quadratic equation will take the form ax2 + bx + c = 0 if all the terms are brought to one side of the equation; and then by Art. 318 the value of x must be either -b+ √(b2-4ac) -b- √(b2 - 4ac) 2α or 2a that is the value of x must be one or the other of two quantities. The result is sometimes obtained thus. If possible let three different quantities a, B, y be roots of the quadratic equation ax2+ bx + c = 0; then, by supposition, aa2+ba + c = 0, aß +bB+c=0, ay2+by+c=0. By subtraction, a (a2 — ẞ2) + b (a − ß) = 0 ; divide by a -ẞ which is, by supposition, not zero; thus this however is impossible, since by supposition a is not zero, and Hence there cannot be three different roots B-y is not zero. to a quadratic equation. 335. In a quadratic equation where the coefficient of the first term is unity and the terms are all on one side, the sum of the roots is equal to the coefficient of the second term with its sign changed, and the product of the roots is equal to the last term. 336. Let a and ẞ denote the roots of the equation the values of expressions in which a and ẞ occur in a symmetrical 337. We have b now put for - and a α a their values in terms of a and ß; thus ax2 + bx + c = a {x23 − (a + ß) x + aß} = a (x − a) (x − ß). Thus the expression ax2 + bx+c is identical with the expression a (x-a) (x-B); that is, the two expressions are equal for all values of x. Hence we can prove the statement of Art. 334 in another manner. For no other value of x besides a and ẞ can make (x − a) (x − ß) vanish; since the product of two quantities cannot vanish if neither of the quantities vanishes. The student may naturally ask if the identity ax2 + bx + c = a (x − a) (x − ẞ) holds in those cases alluded to in Art. 323, where the roots of ax2 + bx + c = 0 are impossible; we shall return to this point in another Chapter. 338. The student must be careful to distinguish between a quadratic equation and a quadratic expression. In the quadratic equation ax2 + bx + c = 0 we must suppose x to have one of two definite values, but when we speak of the quadratic expression ax2 + bx + c, without saying that it is to be equal to zero, we may suppose x to have any value we please. |