Algebra for Beginners with numerous examples . le 2x^ isfound on both sides of the equation, after we have clearedof fractions ; accordingly it can be removed by subtraction,and so the equation remains a quadratic equation. 166 QUADRATIC EQUATIONS. 240. Every quadratic equation can be put in the/orm x^ + px + q = 0, where p a7id q represent some knownnumbers, whgle or fractional, positive or negative. For a quadratic equation, by definition, contains no Eower of the unknown quantity higher than the secondiCt all the terms be brought to one side, and, if necessary,change the signs of all the te
Algebra for Beginners with numerous examples . le 2x^ isfound on both sides of the equation, after we have clearedof fractions ; accordingly it can be removed by subtraction,and so the equation remains a quadratic equation. 166 QUADRATIC EQUATIONS. 240. Every quadratic equation can be put in the/orm x^ + px + q = 0, where p a7id q represent some knownnumbers, whgle or fractional, positive or negative. For a quadratic equation, by definition, contains no Eower of the unknown quantity higher than the secondiCt all the terms be brought to one side, and, if necessary,change the signs of all the terms so that the coeflBcient ofthe square of the unkno^vn quantity may be a positivenumber; then divide every term by this coefiBcient, andthe equation takes the assigned form. For example, suppose 7a:—4.^^=5. Here we have therefore Ax^-lx-^b^O; 7x 5 therefore i>fi—t+i = ^ • 4 4 ^ 7 5 Thus in this example we have |H[|k7 and 5=t. 241. Solve By transposition, add(|y, a^+^^ + (f) Pextract the square root, **? 9 therefore x=-^^ ^l^^^^:^. 242. We have thus obtained a general formuldt^rthe roots of the quadratic equation X+px-^q = 0, namely,that X must be equal to v -y+x/(j?*-4g) -p-J(pi-4q) *^ 2 2 We shall now deduce from this general formrda acmevery important inferences, which will hold for any quad-ratic equation, by Art. 240. QUADRATIC EQUATIONS. 167 243. A quadratic equation cannot have more thantteo roots. For we have seen that the root mitst be one or theother of two assigned expressions. 244. In a quadratic equation where ttie trms areall on one side, and the coefficient of the square of theun/cnotcn quantity is unity, the sum of the roots is eqtuilto the coefficient of the second term with its sign changed,and the product of the roots is equal to the last term. For let the equation be a?- +px + q = 0;the sum of the roots is -P^s/U^-^^) ^ z^lVlti^J^ ,hat is -p; the product of the i-og^is -P + JiW^^) ^ -P- sf{P^-*q) that is ^ ^^ ^^-, that is q. 245. The preceding Arti
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