Transcendental Equations; Roots of Equations.

107. But when an equation does not admit of being reduced to a series of monomials, or, when being so reduced, it contains terms in which the unknown quantities or their powers enter otherwise than as factors, it is said to be transcendental; and the consideration of such equations belongs to the higher branches of mathematics.

Thus,

a* b

(x + a) y + b = c,

are transcendental equations.

108. An equation is said to be solved, when the values of its unknown quantities are obtained; and these values are called the roots of the equation.

109. The reduction and solution of all equations depends upon the self-evident proposition, that

Both members of an equation may be increased, diminished, multiplied, or divided by the same quantity, without destroying the equality.

110. Corollary. If all the terms of an equation have a common factor, this factor may be suppressed.

111. EXAMPLES.

1. If the factor common to the terms of the equation a2 x5 +3 a3 x2 a2 x2

=

is suppressed, what is the resulting equation?

Ans. 233 a = 1.

2. If the factor common to the terms of the equation

is suppressed, what is the resulting equation?

Ans. a3a2 x = 1.

To free an Equation from Fractions.

112. Problem. To free an equation from frac tions.

Solution. Reduce, by arts. 67 and 68, all the terms of the equation to fractions having a common denominator, and suppress the common denominator, prefixing to the numerators the signs of their respective fractions.

Demonstration. For suppressing the denominator of a fraction is the same as multiplying the fraction by its denominator; and, consequently, both the members of this equation are, by the preceding process, multiplied by the common denominator.

113. Corollary. It must be strictly observed that, when the denominator of a fraction is removed, the sign, which precedes the fraction, affects all the terms of the numerator. If therefore this sign is negative, all the signs of the numerator are to be reversed.

Solution. This equation, when its terms are reduced to

To free an Equation from Fractions.

Suppressing the common denominator, we have

or

ad+bc―(a—c) = b d h x — bd,

ad+bc-a+c=bdhx-bd.

2. Free the equation

3a-5x 2α-x

+

=

a + f_dx

a-c

d

a-c

from fractions.

Ans. 3 ad-5dx+2a2-ax-2ac+cx=ad+

Ans. 684-214 x — 14 x2 = 612x+324-240 x2 —

Ans. x2+2xy + y2—x2+2xy—y2=x+y−x+

To free a Fraction from negative Exponents.

115. Corollary. If the given equation contains negative exponents, it can be freed from them by rts. 80 and 82.

117. Theorem.

= a4 x x2 a

- x2 α.

A term may be transposed from one member of an equation to the other member, by merely reversing its sign; that is, it may be suppressed in one member and annexed to the other member with its sign reversed from to, or from to +.

Proof. For suppressing it in the member in which it at first occurs is the same as subtracting it from that member; and annexing it to the other member with its sign reversed is, by art. 26, subtracting it from the other member; and, therefore, by art. 109, the equality is preserved.

118. Corollary. All the terms of an equation may be transposed to either member, leaving zero in the other member; and the polynomial thus formed may be reduced to its simplest form, by arts. 20 and 110.

to its simplest form in a series of monomials.

Solution. This equation, freed from fractions by arts. 112 and 113, is

7x2+1+7x=6x+2—5 x2+1—x”—3 x” — 6 xn+2, which becomes, by the transposition of its terms and by the reduction of art. 20,

12x+1+11 x = 0,

and, by striking out the factor x2,

12x+11= 0.

2. Reduce the equation

x+1

=

x2+1 x-1

x2 - 1 (x+1)2 X- - 1

to its simplest form in a series of monomials.