3. Given,+V, to find x.

151. Every equation which can be reduced to the form

xm + 2px" = १,

in which m and n are positive whole numbers, and 2p and q, known quantities, is called a trinomial equation.

Hence, a trinomial equation contains three kinds of terms: viz., terms which contain the unknown quantity affected with two different exponents, and one or more known terms.

If we suppose m = 2 and n = 1, the equation becomes

x2 + 2px = q,

a trinomial equation of the second degree.

152. The resolution of trinomial equations of the second degree, has already been explained, and the methods which were pursued are, with some slight modifications, applicable to all trinomial equations in which m = 2n, that is, to all equations of the form

x2n + 2pxn = q.

Let us take, as an example, the trinomial equation of the fourth degree,

and by substituting 2p for the co-efficient of x2, and q for the absolute term, we have

If now, we make

x2 + 2px2 = q.

x = ±√y

x2y, and consequently,

we shall have

hence,

y2 + 2py = q, and y=-p±√q+p2:

x=±√-p±√q + p2.

We see that the unknown quantity has four values, since each of the signs + and -, which affect the first radical can be combined in succession with each of the signs which affect the second; but these values taken two and two are numerically equal, and have contrary signs.

Substituting these values, in succession, for y in the equation

Substituting these values, in succession, for y, and we have

1st. x2 8, which gives x = + 2√2, and x =

2d. x2 = -1, which gives x = + √-1, and

The last two values of x are imaginary.

3. Let us take the literal equation

x=

2

2.

153. Before resolving the general case of trinomial equations, it may be well to remark that, the nth root of any quantity, is an expression which multiplied by itself n-1 times will produce the given quantity.

The method of finding the nth root has not yet been explained, but it is sufficient for our present purpose that we are able to indicate it.

Let it be required to find the values of y in the equation y2n + 2pyn = q.

If we make y" = x, we have y2n = = x2, and hence, the given equation becomes

If we suppose n = 2, the given equation becomes a trinomial

equation of the fourth degree, and we have

y=√-p±√q + p2.

154. The resolution of trinomial equations of the fourth degree, therefore, gives rise to a new species of algebraic operation : viz., the extraction of the square root of a quantity of the form

in which a and b are numerical or algebraic quantities.

To illustrate the transformations which may be effected in ex

pressions of this form, let us take the expression 3 ± 5. By squaring it we have

(3± 5)2 = 9 ± 65+5=146√5:

hence, reciprocally, √ 14 ± 6√5 = 3√5.

As a second example, we have

(√7±√11)2 = 7 + 2√77 + 11 = 18 ± 2√77 :

hence, reciprocally,

18277=√7 ± √11.

Hence we see, that an expression of the form

and when this transformation is possible, it is advantageous to effect it, since in this case we have only to extract two simple square roots; whereas, the expression

requires the extraction of the square root of the square root.

155. If we represent two indeterminate quantities by p and q, we can always attribute to them such values as to satisfy the equations

Now, if p and q are irrational monomials involving only single radicals of the second degree, or, if only one is irrational, it follows that p2 and q2 will be rational; in which case, p2 - q2, or its value, a2- b, is necessarily a rational quantity, and consequently, a2-b is a perfect square.

Under this supposition, a transformation can always be effected that will simplify the expression.

By squaring equations (1) and (2), we have

p2 + 2pq + q2 = a +√b

If we denote the second member of equation (3) by c, we shall

have

p2-q2 = c

(5).

By adding the two last equations and subtracting equation (5) from (4), we have