2x2-5x+6+10/2x2-5x+6=39,

(2x2 - 5x+6) + 10 √2x2 − 5 x + 6 + 25 = 25 + 39 = 64,

2x2-5x+6=9, or 169;

which leaves two common quadratics for solution.

211.

When there are more equations and unknown quantities than one, a single equation involving only one of the unknown quantities may sometimes be obtained by the rules laid down for the solution of simple equations; and one of the unknown quantities being discovered, the others may be obtained by substituting its value in the other equations.

From the 2nd equation, xy + 2y − x − 3y = x + 2,

by substitution, (8 − y) y − 2 (8 − y) − y = 2,

2

8y-y2 - 16+ 2y − y = 2,

The solution of equations will often be rendered more simple by particular artifices, the proper application of which is best learned by experience*.

* Many of these artifices are pointed out in the Appendix.

212. It may sometimes be of use to substitute for one of the unknown quantities the product of the other and a third unknown quantity.

This substitution may be successfully applied whenever the sum of the dimensions (Art. 63) of the unknown quantities in every term of each equation is the same.

213. The operation may sometimes be facilitated by substituting for the unknown quantities the sum and difference of two others This artifice may be used, when the unknown quantities in each equation are similarly involved.

OBS. In algebraical analysis it is frequently useful to observe whether the algebraical expressions under consideration are homogeneous or not, that is, whether the dimensions of every term be the same or not; for, if this homogeneity be found at first, no legitimate operation can destroy it; or, if it be not found at first, it cannot be introduced; and thus an easy test is afforded, to a certain extent, of the accuracy of each succeeding step in the analysis.

PROBLEMS PRODUCING QUADRATIC EQUATIONS.

For example, if the equation

a x2+ b2x + c3 = 0,

125

be proposed for solution, in which every term is of three dimensions, that is, which is homogeneous, every step in the process will present an homogeneous equation, if it be correct.

As a simple case it may be well to observe that, if the proposed equation be homogeneous, the final result must be so. A proper attention to this observation will frequently detect an error in the solving an equation.

process of

PROBLEMS PRODUCING QUADRATIC EQUATIONS.

214. PROB. I. A person bought a certain number of oxen for 80 guineas, and if he had bought 4 more for the same sum, they would have cost a guinea a piece less; required the number of oxen and price of each.

.. x = ± 18 −2=16, or - 20, numb. of oxen ;

5 guineas, the price of each.

In this, and in many other cases, especially in the solution of philosophical questions, we deduce from the algebraical process answers which do not correspond with the conditions. The reason seems to be, that the algebraical expression is more general than the common language; and the equation, which is a proper representation of the conditions, will also express other conditions, and