Elements of Algebra

566 ADDITIONS. CHAP. VII. $b^2 - 4ac = A$, $bd - 2ae = g$, $d^2 - 4af = h$, and $As^2 + 2gs + h$ must be a square; representing this square by $y^2$, in order that we may have the equation, $As^2 + 2gs + h = y^2$, and taking the value of $y$, we shall have $As + g = \sqrt{(Ay^2 + g^2 - Ah)}$; so that we shall only have to make a square of the formula, $Ay^2 + g^2 - Ah$. If, therefore, we also make $g^2 - Ah = B$, we shall have to render rational the radical quantity, $\sqrt{(Ay^2 + B)}$; which we may do by the known methods. Let $\sqrt{(Ay^2 + B)} = x$, so that the equation to be resolved may be $Ay^2 + B = x^2$; we shall then have $As + g = \pm x$. Now, we already have $2ar + bs + d = \pm y$; so that, when we have found the values of $x$ and $y$, we shall have those of $r$ and $s$, by the two equations, $s = \frac{\pm x - g}{A}$, $r = \frac{\pm y - d - bs}{2a}$ Now, as $r$ and $s$ must be whole numbers, it is evident, 1st, that $x$ and $y$ must be whole numbers likewise; 2dly, that $\pm x - g$ must be divisible by $A$, and $\pm y - d - bs$ by $2a$. Thus, after having found all the possible values of $x$ and $y$, in whole numbers, it will still remain to find those among them that will render $r$ and $s$ whole numbers. If $A$ is a negative number, or a positive square number, we have seen that the number of possible solutions in whole numbers is always limited; so that in these cases, we shall only have to try, successively, for $x$ and $y$, the values found; and if we meet with none that give whole numbers for $r$ and $s$, we conclude that the proposed equation admits of no solution of this kind. There is no difficulty, therefore, but in the case of $A$ being a positive number, not a square; in which we have seen, that the number of possible solutions in whole numbers may be infinite. In this case, as we should have an infinite number of values to try, we could never judge of the solvibility of the proposed equation, without having a rule, by which the trial may be reduced within certain limits. This we shall now investigate. 77. Since we have (Art. 65), $x = ny - bz$, and (Art. 72), $y = pt - (bq - np)u$, and $z = qt + (cp - nq)u$, it is easy to perceive, that the general expressions of $r$ and $s$ will take this form,