New higher algebra; an analytical course designed for high schools, academies, and colleges

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. 1873 Excerpt: ...+ /9 + f! and-y/y+f is +( + £); consequently, Every quadratic equation may have two roots. 1. Given 2a? + 12a; + 36 = 356, to find the values of x. OPERATION. 2a+ 12a; + 36 = 356 By transposition, 2x2 + 12 a; = 320 Dividing by 2, x2 + 6 a; = 160 Completing the square, a;2 + 6 a; + 9 = 169 Extracting the square root, x + 3 = ±13 Transposing, x =--3 ± 13 Whence, x = 10, or--16. Here, taking 13, the positive root of 169, we find x = 10; but taking--13, the negative root, we find x =--16. 2. Given a;2 77-12 a; + 30 = 3, to find the values of x. OPERATION. x2--12 x +30 = 3 By transposition, x2--12 a; =--27 Completing the square, x2--12 a; + 36 = 9 Extracting the square root, x--6 = ±3 Transposing, x = 6 ± 3 Whence, x = 9, or 3; where both values of x are positive. 3. Given--3x2--1 x = Js0-, to find the values of x. OpERATION.--3 a;2--7x = JtfDividing by--3, x2-j-x =--Completing the square, x2--£ x--=.£% Extracting the square root, x-f-£ = ± f Transposing, x =--£ ± $ Whence, a; =--§, or--§; where both values of x are negative. The results obtained in each of these operations may be readily verified. Hence the following General Rule. Reduce the given equation to the form x2--p x = q. Complete the square by adding the square of half the coefficient of x to both members. Extract the square root of both members, and solve the simple equation thus produced. Sometimes the first, member of the equation reduces to a perfect square, and the second to 0. Then the root may be found directly; but x will really have but one value. Thus, x2--6 x =--9 by transposition becomes x2--6z + 9 = 0, where the first member is the square of 1--3, whence x = 3 ± 0 = 3. It is, however, convenient in this case to consider...