Elements of Algebra

218 ELEMENTS SECT. IV. which reason we shall first consider those, in which one of the terms is wanting. Now, if the term $ax^2$ were not found in the equation, it would not be a quadratic, but would belong to those of which we have already treated; and if the term, which contains only known numbers, were wanting, the equation would have this form, $ax^2 \pm bx = 0$, which being divisible by $x$, may be reduced to $ax \pm b = 0$, which is likewise a simple equation, and belongs not to the present class. 629. But when the middle term, which contains the first power of $x$, is wanting, the equation assumes this form, $ax^2 + c = 0$, or $ax^2 = \mp c$; as the sign of $c$ may be either positive, or negative. We shall call such an equation a *pure equation of the second degree*, and the resolution of it is attended with no difficulty; for we have only to divide by $a$, which gives $x^2 = \frac{c}{a}$; and taking the square root of both sides, we find $x = \sqrt{\frac{c}{a}}$; by which means the equation is resolved. 630. But there are three cases to be considered here. In the first, when $\frac{c}{a}$ is a square number (of which we can therefore really assign the root) we obtain for the value of $x$ a rational number, which may be either integral, or fractional. For example, the equation $x^2 = 144$, gives $x = 12$. And $x^2 = \frac{9}{16}$, gives $x = \frac{3}{4}$. The second case is, when $\frac{c}{a}$ is not a square, in which case we must therefore be contented with the sign $\sqrt{}$. If, for example, $x^2 = 12$, we have $x = \sqrt{12}$, the value of which may be determined by approximation, as we have already shewn. The third case is that, in which $\frac{c}{a}$ becomes a negative number: the value of $x$ is then altogether impossible and imaginary; and this result proves that the question, which leads to such an equation, is in itself impossible. 631. We shall also observe, before proceeding farther, that whenever it is required to extract the square root of a number, that root, as we have already remarked, has always two values, the one positive and the other negative. Suppose, for example, we have the equation $x^2 = 49$, the value