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• If the former values of a -y, namely, + A/116 or + 10.77 be taken, then for the affirmative values a = | 1.385, &c. and y =.615, &c. and for the negative values a =.615, &c. and y = 11.385, &c. both of which values answer the conditions of the question equally with those given above. It appears from the solution, that this example (which was inserted by mistake) is not an adsected, but a pure quadratic, and therefore is misplaced; the same may be said of the thirteenth example, **
98. By this rule may be solved all equations whatever, wherein there are only two different dimensions of the unknown quantity, provided the index of the one be exactly double that of the other.
Rule. Having completed the square and extracted the root as before, transpose the known quantities which are on the same side with the unknown one, and then extract the root implied by the index of the unknown quantity, from both sides of the equation ".
p Every equation will have as many roots as the unknown quantity has dimensions; thus, in the 25th example, r being in the fourth power, the equation will have four roots, as appears by the solution. In example 26, y comes out equal to 49 or 25, the latter of which being substituted in the equation, will not answer, except —5 be substituted for the square root instead of +5, the reason of which is obvious, since the 25 arose from –5 x —5. One root only is required in the following examples, as finding the rest would, in many eases, require rules which have not yet been given.
Every problem proposed to be solved algebraically, contains some conditions laid down, which are called the data; from whence one or more quantities are required to be found, called the quasita. The first thin of necessary to be done preparatory to the solution, is to understand clearly the import and signification of the problem: it must be freed from every thing ambiguous and un
* The word problem is derived from the Greek weedanaz. An algebraic problem is a proposition wherein some unknown truth is required to be investigated or discovered, and the truth of the discovery demonstrated.
necessary; the conditions, and the manner of their dependance on each other, must be clearly ascertained and stated, and they must be carefully distinguished from the quantities proposed to be found: when this is accomplished, the conditions of the proposed problem will be exhibited under the form of one or more equations; namely, as many equations as there are unknown quantities, the solution of which is the subject of the preceding rules. Much depends on a proper substitution for the quantities required: no general rule for this can be given; sometimes a letter must be put for each; frequently, having substituted a letter for one of the unknown quantities, expressions for the others may be derived by means of this and the conditions proposed, without the aid of new letters; sometimes a substitution for the sum, difference, product, quotient, roots, powers, &c. of the unknown quantities, may be conveniently made; but the proper application of these must be learned by experience and practice. The following modes of substitution will apply in many cases. For one unknown quantity put a, for two put r and y, a being the greater, y the less; for their sum a 4-y, for their difference r—y, for the square of the greater wo, for the cube root of the less * Vy, for the sum of their squares ao-Hy”, for the difference of their squares wo—y”, for the square of the sum r-Fyl”, for the cube root of their difference "yo-y, for their product ry, their
quotient #. where the greater is proposed to be divided by the
less, or #, where the less is proposed to be divided by the greater.
In general, the sum of any two quantities is represented by interposing the sign + between them; the difference by the sign –, the product by the sign x, or by placing them as coefficients to each other, and the quotient by placing the dividend above the divisor; following in every case the method applicable to it, as proposed in algebraic notation. 1. What number is that, to which 9 being added, the sum will be ‘23? Let the required number be represented by r. To which adding 9, the sum will be a +9. This, by the problem, equals 23, whence r +9–23.