This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1887 Excerpt: ...ax2 ] bx + c = 0 has two equal roots. The condition 2--4 a c = 0 is equally the condition that the expression ax2 + bx + c shall be an exact square, and that the two roots of the equation ax2 + bx + c = 0 shall be equal. In many applications of Algebra to Geometry and Physics the student will discover the importance of ...
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1887 Excerpt: ...ax2 ] bx + c = 0 has two equal roots. The condition 2--4 a c = 0 is equally the condition that the expression ax2 + bx + c shall be an exact square, and that the two roots of the equation ax2 + bx + c = 0 shall be equal. In many applications of Algebra to Geometry and Physics the student will discover the importance of the difference between the statements that an equation has only one root, and that an equation has two equal roots. 422. From the last few articles it is clear that if xl, x2 be the roots of the equation ax2 + bx + c = 0, the expression ax2 + bx + c is identically equal to a(xx1) (x--x2). For, in Art. 418 it is shown that if b2--4 ac is positive, aaP + bx+c = Vax+ 5. Vax +--+ I 2a 2a ) I 2a 2a i or, multiplying out, xl +x2) x + xlx2--0. Thus the quadratic equation whose roots are 1 and 3 is (x-l)(x-3) = 0, or x2--4x + 3 = 0. This last result can be compared with Art. 404. 425. The relations (l) and (2) of Art. 414, namely, b C X, x0 =-1 i a which can also be deduced from the identity of Art. 422, sometimes simplify the solution of a given quadratic equation. Thus the equation + 2001-2002 = 0 is obviously satisfied by the value x = 1. Hence one root being unity and the product of the two roots being--2002, it follows that the other root must be-2002. The two roots are thus 1, -2002. 426. The same relations can be used to express any symmetrical function of the two roots in terms of the coefficients of the given equation. A symmetrical function means an expression involving the two quantities in such a manner that its value is not altered when they are interchanged. Thus xl + x., x + x.f, i3x-i + xx-i are instances of symmetrical functions of xx and x2. 427. Since a + x., =-squaring these equal quantities it follows that xl + 2r1ra + r=-2, ..
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