Elementary Algebra for the Use of Preparatory Schools (Paperback)

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. 1894 Excerpt: ...and since we can write down the square root of any trimonial expression which is a complete square, it follows that the square root of any expression which is a complete square can be written down by inspection, provided that the expression only contains two different powers of some particular letter. For example, to find the square root of a2 + 62 + c2 + 2 6c + 2 ac + 2 ab. Arranging the expression according to the descending powers of a, we have a1 + 2 a(b + c) + (ft2 + 2 be + c2), that is, a2 + 2 a(b + c) + (6 + c)2, which is a + (6 + c)2. Consider, for example, a? + 2 xy + 3f (L), whose square is x + ix"y + l()xY + 12xys + 9yi... (ii.), both expressions being arranged according to descending powers of x. We may write the square of x2 + 2 xy + 3 y2 in either of the following forms: x2 + 2xy + 3y2)2=a? + 2x22xy + 3y2) + 2xy + 3f)2 (iii.), (a;2 + 2 xy) + 3y22 = (as + 2xy)2 + 2x + 2 xy)3y + (3.V2)2 (iv.). Now it is clear that the first term of (ii.) is the square of the first term of (i.). Hence the first term of the root of (ii.) is found by taking the square root of its first term. Again, we see from (iii.) that when we have subtracted x4 (the square of the first term of the root), the term in the remainder which contains the highest power of a; is 2 a;2 x 2 xy, which is twice the product of the first and second terms of the root. Hence, after subtracting from (ii.) the square of the first term of the root, the second term is obtained by dividing the first term of the remainder by twice the first term of the root. Again, we see from (iv.) that when we have subtracted (x2--2xyY, that is, the square of the part of the root already found, the term in the remainder which contains the highest power of x is 2xF x Sy2, which is twice the product of the firs...