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The application of the rules of this chapter to the resolving of equations, shall be explained in the fucceeding chapters, which treat of the folution of the different claffes of them; but fome examples of their use in preparing equations for a folution, are the following.
If a member of an equation be a furd root, then the equation may be freed from any furd, by bringing that member first to ftand alone upon one fide of the equation; and then taking away the radical fign from it, and raising the other fide to the power denominated by the index of that surd.
This operation becomes a neceffary step towards the solution of an equation, when any of the unknown quantities are under the radical fign.
Example. If 3x2—a2+2y=a+y
If the unknown quantity be found only under the radical fign, and only of the first dimenfion, the equation will become fimple, and may be refolved by the preceding rules. jordings to griviobɔra
Thus, if 4+16+596
Then /4x+164 sila to tea
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If the unknown quantity in a final equation has fractional exponents, by means of the preceding rules a new equation may be fubftituted, in which the exponents of the unknown quantity are integers.
Thus, if x2+3×3=10, by reducing the
furds to the fame denomination, it becomes x2+3x=10; and if x=x, then '+3x+ =10; and if this equation be refolved, N
from a value of ༧, a value of x may be got by the rules of the next chapter. Thus al
-3×3=100. If x=
so, if x+2x3—3x3
If x=x, this
equation becomes x+2x3-3x2=100.
In general, if x1+x"=a, by reducing
the furds to the fame denomination, x"9+
=a, and if x=x, then the equation
is +x=a, in which the exponents of
≈ are integers; and ≈ being found, x is to
be found from the equation "%,
EQUATIONS were divided into orders
according to the highest index of the unknown quantity in any term. (Chap. III.).
Equations are either pure or adfected.
Def. I. A pure equation is that in which only one power of the unknown quantity is found.
2. An adfected equation is that in which different powers of the unknown quantity are found in the feveral terms.
Thus, a2+ax2=b3‚ax2—b2=m2+x2, arë pure equations.
* And x2-ax=b2, x3 +x2=17, are adfected.
I. Solution of Pure Equations.
Rule. Make the power of the unknown quantity to ftand alone, by the rules formerly given, and then extract the root, of the fame denomination, out of both fides, which will give the value of the unknown quantity.
The index of the power may also be fractional; as, in the last example, m may be any number whatever. Let m=1, then, as before,