The elements of algebra

1st.

Suppose the number to be 20, (8)

then 20 + 10 = 30,

but 3 × 20 = 60,

... e =

- 30.

2dly. Suppose the number to be (30), (s′)

XXV. Since every quadratic equation may be reduced to the form a2 + px + q = 0, in which p and q may be positive or negative, we assume this as the general equation including every other.

which are the only two values of a that will satisfy the equa

tion.

Now, from this result, it follows,

1st. That there is no possible value of x, if p2 <4q.

2dly. That the values of x are equal to each other, and -2, if p2 = 4q.

each equal to

3dly. That there are two unequal values of x, whose sum isp, if p2 > 4q.

Again, it appears that

1st. If q be negative, since p2 - 4q will then be greater than p and always possible, there can be but one positive value of x.

2dly. If p be negative, q positive, and p2>4q, there will be two positive values of x.

200.

The last two conclusions may also be deduced from Art.

COR. Similar conclusions may be drawn with regard to b

the equation ax2 + bx + c = 0, by substituting for

and

for q. Thus, if a proposed equation be of the form bx-ax2=c, in which a, b, c are positive quantities, there will be two positive values of x, when 4ac < b2; and no value of x at all when 4ac > b2.

INDETERMINATE COEFFICIENTS.

XXVI. It may be objected to the proof in Art. 324, that it is rather assumed than proved that every equation has only a certain number of values of a which will satisfy it; and that it does not include those cases in which the assumed series is an "infinite series." No such objections can be made to the following proof:

Let A + B + C x2 + ...= a + bx + cx2 +... be an identical equation, that is, hold for any value whatever of a; then

= 0,

A ~ a + (B ~ b) x + (C ~ c) x2 + and, if A~a is not equal to 0, let it be equal to some quantity p; then we have

= p.

(B ~ b) x + (C ~ c) x2 + &c. And since A, a, are invariable quantities, their difference p must be invariable; but p = (B ~ b) x + (C ~ c) x2 + .... a quantity which may have various values by the variation of x; that is, we have the same quantity (p) proved to be both fixed and variable, which is absurd. Therefore there is no

quantity (p) which can express the difference A~ a, or, in other words,

Therefore, by what has been proved, B = b. And so on, for the remaining coefficients of like powers of x.

for all values whatever of x and y, then the coefficients of like quantities are equal to each other, that is,

Aa, Bb, C=c, A'a', B'b', A" a", &c.

Since may receive any value whatever, suppose it to have some fixed value while y is variable, then the equation may be put under the form

where A, B, C,, &c. a, b, c,, &c. are invariable coefficients, and

Now, by Art. xxvi, A‚= a,, B‚=b,, C, = c,, &c. ; therefore

Then by Art. xxvI, again, since a may have any value whatever,

Aa, Bb, C=c, A' a', B'b', A" = a", &c.

SUMMATION OF SERIES.

XXVIII. Series are sometimes proposed for summation which are not actually composed of terms in Arithmetical or Geometrical Progression, but which may be made so by arrangement.

Thus,

Let the sum of n terms of the following series be required, 1 + 5 + 13 + 29 +61 + &c.