Elements of Algebra

244 ELEMENTS SECT. IV. CHAP. IX. *Of the Nature of Equations of the Second Degree.* 689. What we have already said sufficiently shews, that equations of the second degree admit of two solutions; and this property ought to be examined in every point of view, because the nature of equations of a higher degree will be very much illustrated by such an examination. We shall therefore retrace, with more attention, the reasons which render an equation of the second degree capable of a double solution; since they undoubtedly will exhibit an essential property of those equations. 690. We have already seen, indeed, that this double solution arises from the circumstance that the square root of any number may be taken either positively, or negatively; but, as this principle will not easily apply to equations of higher degrees, it may be proper to illustrate it by a distinct analysis. Taking, therefore, for an example, the quadratic equation, $x^2 = 12x - 35$, we shall give a new reason for this equation being resolvible in two ways, by admitting for $x$ the values 5 and 7, both of which will satisfy the terms of the equation. 691. For this purpose it is most convenient to begin with transposing the terms of the equation, so that one of the sides may become 0; the above equation consequently takes the form $x^2 - 12x + 35 = 0$; and it is now required to find a number such, that, if we substitute it for $x$, the quantity $x^2 - 12x + 35$ may be really equal to nothing; after which, we shall have to shew how this may be done in two different ways. 692. Now, the whole of this consists in clearly shewing, that a quantity of the form $x^2 - 12x + 35$ may be considered as the product of two factors. Thus, in reality, the quantity of which we speak is composed of the two factors $(x-5) \times (x-7)$; and since the above quantity must become 0, we must also have the product $(x-5) \times (x-7) = 0$; but a product, of whatever number of factors it is composed, becomes equal to 0, only when one of those factors is reduced to 0. This is a fundamental principle, to which we must pay particular attention, especially when equations of higher degrees are treated of. 693. It is therefore easily understood, that the product