QUADRATIC EQUATIONS. 200. When the terms of an equation involve the square of the unknown quantity, but the first power does not appear, the value of the square is obtained by the preceding rules*; and by extracting the square root on both sides, the quantity itself is found. Ex. 1. 5 x2 - 45 = 0; to find x. and The signs are both prefixed to the root, because the square root of a quantity may be either positive or negative (Art. 147). The sign of a may also be negative; but still will be either equal to + 3 or - st. Ex. 2. aa2bcd; to find x. 00 201. If both the first and second powers of the unknown quantity be found in an equation, arrange the terms according to the dimensions of the unknown quantity, beginning with the highest, and transpose the known quantities to the other side; then, if the square of the unknown quantity be affected with a coefficient, divide all the terms by this coefficient, and if its sign be negative, change It is obvious that the rules proved in Arts. 186-192, apply to all equations, quadratic, cubic, &c. as well as simple, because they are founded simply upon the Axioms (Arts. 79-82.) †This may be shewn as follows:-suppose x2=a2, then extracting the square root of both sides, since √x2=±x, and Va3±a, we have = But it is evident that (1) and (4) are in fact the same equation, and also (2) and (3); so that a = a includes all the four equations. the signs of all the terms (Art. 187), that the equation may be reduced to this form, apa = ±q. Then add to both sides the square of half the coefficient of the first power of the unknown quantity, by which means the first side of the equation is made a complete square, (Art. 153), and the other consists of known quantities; and by extracting the square root of both sides, a simple equation is obtained, from which the value of the unknown quantity may be found. Ex. 1. Let x2 + px =q; now, we know that x2 + px + 2 p2 4 p2 is the square of x + and we have Р (Art. 153); add therefore to both sides, then by extracting the square root on both sides, x2 + px + p2 4 By transposition a2 - 12x=-35; and adding the square of or 6 to both sides of the equation, x2 12x+36=36 35 = 1 then extracting the square root of both sides, x = 6 ± 1 = 12 2 7 or 5; either of which, substituted for x in the original equation, answers the condition, that is, makes the whole equal to nothing. By this process two values of x are found; but on trial it appears, that 18 does not answer the conditions of the equation, if we suppose that 5x + 10 represents the positive square root of 50+ 10. The reason is, that 5 + 10 is the square of −√5x + 10 as well as of + √5x+10; thus by squaring both sides of the equation 5x+10=8x, a new condition is introduced, and a new value of the unknown quantity corresponding to it, which had no place before. Here 18 is the value which corresponds to the supposition that It should be particularly observed, that since + ≈ × + y is equal toxxy, in the multiplication and involution of quantities new values are always introduced, which, if not again excluded by the nature of the question, will appear in the final equation. 203. If a quadratic equation appear under the form ax2+ bx= ±C, the left hand side may be made a complete square, and the equation solved, by another method, as follows:: Multiply the whole equation by 4a, that is, four times the coefficient of x2, then we have add b2, the square of the extract the 4a2x2+4abx = ± 4ac; coefficient of x, then 4a2x2± 4abx+b2 = b2± 4a c square root, 2ax ± b = ± √b2± 4ac ; adding 5o, or 25, 36x2 – 60 x + 25 = 24 + 25 = 49; Ex. 2. acx2-bcx+ adx=bd; to find x. Here acx2 - (bc-ad) x = bd; multiply by 4ac, 4a2c2x2-4ac (bc-ad) x = 4abcd, Extract square root, 2acx-(bc-ad) = ± (bc+ad); 2acx=bc-ad± (bc+ad) =2bc, or -2ad; 204. A quadratic equation has no more than two distinct values of the unknown quantity which will satisfy it. For, if possible, let the equation ax + bx + c = 0 have three distinct values of x, viz. a, ß, y. Subtracting (2) from (1), a (a2 - ẞ2) + b (a − ẞ) = 0; .. a (a +ß) + b = 0 ....................... (i.) Subtracting (ii.) from (i.) a (B − y) = 0. But a is not equal to 0, for otherwise the proposed equation would not be a quadratic equation; Hence a quadratic equation has not three distinct values of x, but it may have two. 205. In any quadratic equation of the form x2 + px + q = 0, − p = the sum of the two values of x, q= their product. and = |