Or, if the second and last terms be taken either positively or negatively, as they may happen to be, the general equation ac2+bac=-Ec, or a:* +}=+. which comprehends all the three cases above mentioned, may be resolved by means of the following rule : RULE. Transpose all the terms that involve the unknown quantity to one side of the equation, and the known terms to the other; observing to arrange them so, that the term which contains the square of the unknown quantity may be positive, and stand first in the equation. Then, if this square has any coefficient prefixed to it, let all the rest of the terms be divided by it, and the equation will be brought to one of the three forms abovementioned. In which case, the value of the unknown quantity c is always equal to half the coefficient, or multiplier of z, in the second term of the equation, taken with a contrary sign, together with + the square root of the square of this number and the known quantity that forms the absolute, or third term, of the equation. (c) (c) This rule, which is more commodious in its practical application, than that usually given, is founded upon the same principle; being derived from the well known property. that in any quadratic *2 =E air--- b, if the square of half the coefficient a L Note. All equations, which have the index of the unknown quantity, in one of their terms, just double that of the other, are resolved like quadratics, by first finding the value of the square root of the first term, according to the method used in the above rule, and then taking such a root, or power of the result, as is denoted by the reduced index of the unknown quantity. Thus, if there be taken any general equation of this kind, as 22m+ax”=b, we shall have, by taking the square root of a ", and observing the latter part of the rule, of the second term of the equation be added to each of its sides, so as to render it of the form that side which contains the unknown quantity will then be a complete square ; and, consequently, by extracting the root of each side, we shall have which is the same as the rule, taking a and b in + or – as they may happen to be. It may here, also, be observed, that the ambiguous sign +, which denotes both + and —, is prefixed to the radical part of the value of r in every expression of this kind, because the Square roof of any positive quantity, as a2, is either +a or—a; for(+a) X (+a), or (- a)x( — a) are each- + as : but the square root of a negative quantity, as – as, is imaginary, or unassignable there being no quantity, either positive or negative, that, when multiplied by itself, will give a negative product To this.we may also further add, that from the constant OCCurrence of the double sign before the radical part of the above expression, it necessarily follows, that every quadratic equation mu have two roots ; which are either both real, or both imaginary, according to the nature of the question. |