Elements of Algebra

CHAP. V. OF ALGEBRA. 217 and then reduced to a single term. Let us take, for example, the equation, $2x^2 - 3x + 4 = 5x^2 - 8x + 11$; we first subtract $2x^2$, and there remains $-3x + 4 = 3x^2 - 8x + 11$; then adding $8x$, we obtain, $5x + 4 = 3x^2 + 11$; lastly, subtracting 11, there remains $3x^2 = 5x - 7$. 626. We may also bring all the terms to one side of the sign $=$, so as to leave zero, or $0$, on the other; but it must be remembered, that when terms are transposed from one side to the other, their signs must be changed. Thus, the above equation will assume this form, $3x^2 - 5x + 7 = 0$; and, for this reason also, the following general formula represents all equations of the second degree; $ax^2 \pm bx \pm c = 0$; in which the sign $\pm$ is read plus or minus, and indicates, that such terms as it stands before may be sometimes positive, and sometimes negative. 627. Whatever therefore be the original form of a quadratic equation, it may always be reduced to this formula of three terms. If we have, for example, the equation $\frac{ax+b}{cx+d} = \frac{ex+f}{gx+h}$ we may, first, destroy the fractions; multiplying, for this purpose, by $cx + d$, which gives $ax + b = \frac{cex^2+cfx+edx+fd}{gx+h}$, and then by $gx+h$, we have $agx^2 + bgx + ahx + bh = cex^2 + cfx + edx + fd$, which is an equation of the second degree, reducible to the three following terms, which we shall transpose by arranging them in the usual manner: $\left. \begin{array}{r} ag \\ -ce \end{array} \right\} x^2 + \left\{ \begin{array}{l} +bg \\ +ah \\ -cf \\ -ed \end{array} \right\} x + \left\{ \begin{array}{l} +bh \\ -fd \end{array} \right\} = 0.$ We may exhibit this equation also in the following form, which is still more clear: $(ag - ce)x^2 + (bg + ah - cf - ed)x + bh - fd = 0.$ 628. Equations of the second degree, in which all the three kinds of terms are found, are called complete, and the resolution of them is attended with greater difficulties; for