Elements of Algebra

CHAP. V. OF ALGEBRA. 335 numerous, it will be worth while to fix on some characters, by which their impossibility may be perceived, in order that we may be often saved the trouble of useless trials; which shall form the subject of the following chapter*. *** CHAP. V. *Of the Cases in which the Formula $a + bx + cx^2$ can never become a Square.* 63. As our general formula is composed of three terms, we shall observe, in the first place, that it may always be transformed into another, in which the middle term is wanting. This is done by supposing $x = \frac{y-b}{2c}$; which substitution changes the formula into $a + \frac{by-b^2}{2c} + \frac{y^2-2by+b^2}{4c}$; or $\frac{4ac-b^2+y^2}{4c}$; and since this must be a square, let us make it equal to $\frac{z^2}{4}$, we shall then have $4ac - b^2 + y^2 = \frac{4cz^2}{4} = cz^2$; and, consequently, $y^2 = cz^2 + b^2 - 4ac$. Whenever, therefore, our formula is a square, this last $cz^2 + b^2 - 4ac$ will be so likewise; and reciprocally, if this be a square, the proposed formula will be a square also. If therefore we write $t$, instead of $b^2-4ac$, the whole will be reduced to determining whether a quantity of the form $cz^2 + t$ can become a square or not. And as this formula consists only of two terms, it is certainly much easier to judge from that whether it be possible or not; but in any further inquiry we must be guided by the nature of the given numbers $c$ and $t$. 64. It is evident that if $t = 0$, the formula $cz^2$ can become a square only when $c$ is a square; for the quotient arising from the division of a square by another square being likewise a square, the quantity $cz^2$ cannot be a square, unless * See the Appendix, Ch. V. p. 537, of the Additions by De la Grange.