And if the same roots be extracted on each side, the results are equal:

2

Thus, if x2 = 49, x = 7; if x3 = a3 b3, then x = ab; if x2 + 2x + 1 = y-y+, then x+1=y-, and if x2 - 4 ax + 4a2 = y2+6 by +962, then x 2a = y+3b.

For (13 and 14) when equal quantities on each side of an equation are multiplied or divided by equal quantities, the results will be equal.

COR. Hence, if that side of the equation which contains the unknown quantity be a perfect square, cube, or other power, by extracting the square root, cube root, &c. of both sides, the equation will be reduced to one of lower dimensions :

Thus, if x2+8x + 16 = 36, x + 4 = 6,

if x33x2+3x+1=27, x+1=3,

3

2

if x2 + 2x3 + x2 = 100, x2 + x = 10.

(20.) Any equation may be cleared of a single radical quantity by transposing all the other terms to the contrary side, and raising each side to the power denominated by the surd. If there are more than one surd, the operation must be repeated.

Thus, if x = ax + b2, by squaring each side r2 = ax + b2, which is free from surds.

Also, if √√√x2+7+x=7,

then (17) by transposition, √√x2 +7=7 — x ;

2

and (19) by squaring each side, x2+7=49-14 x + x2, which is free from surds.

Also, if x + √ax = b,

3

then (17) by transposition, ax = b-x;

and (19) by cubing each side, a2x = b3 − 3 b2 x + 3 b x2 — x3, which is free from surds.

2

Also, if √x2+√x2 +21−1=x,

then (17) by transposition, √x+√x2+ 21 = x + 1,

2

and (19) by squaring each side, x2 + √√x2+21=x2+2x+1;

..(17. Cor. 3.)√x2+21=2x+1,

and (19) by squaring each side, x2 + 21 = 4x2 + 4x + 1, which is free from surds.

3

And, if √a3x + √a3 x3 = c,

(19) by cubing each side, a2x + √a3 x3 = c3,

and (17) by transposition a3 3 — a2x;

3

3

=

. (19) by squaring each side, a3x3 c6 - 2a c3 x + ax", which is free from surds.

(21.) Any proportion may be converted into an equation; for the product of the extremes is equal to the product of the means.

Let abc d, by the nature of proportion

.. (18. Cor. 1.) ad = bc.

a

C

(22.) EXAMPLES in which the preceding Rules are applied, in the Solution of Equations.

1. Given 4x + 36=5x+34, to find the value of x. (17) By transposition, 3634 = 5x — 4x,

2. Given x

+ to find the value of x.

5

3

Here 15, the product of 3 and 5, being their least common multiple, every term must be multiplied by it (18. Cor. 1.), and 15 x

105 = 3x + 5 x ;

=

.. (17) by transposition, 15 x 3 x 5 x 105,

(18. Cor. 2.) dividing every term by a, 3x-4b= 2 x − 6 c ; .. (17) by transposition, 3x-2x=4b-6c,

or x=4b-6c.

2

4. Given 3x2-10x=8x+x2, to find the value of x. (18. Cor. 2.) dividing every term by x, 3x

108+ x; .. (17) by transposition, 3 x x = 8 + 10,

or 2x = 18;

.. (18. Cor. 2.) x =

Here 12 is the least common multiple of 2, 3, and 4; (18. Cor. 1.) multiplying both sides of the equation therefore by 12, 6x + 4x = 3 x + 84;

..(17) by transposition, 6x+4x-3x= 84,

(18. Cor. 1.) multiplying by 20, the least common multiple of 4 and 5,

. (17) by transposition, 5x + 120x + 4x = 1136 + 25,

(18. Cor. 1.) multiplying by 6, the least common mul

tiple of 2 and 3,

(7.) An Adfected Quadratic is one which involves the square of the unknown quantity, and also the simple power and known quantities.

Thus, a x2 + b = 0 is a pure quadratic,

and ax2 + bx + c = 0 is an adfected quadratic.

(8.) The Resolution of Equations is the determining, from some quantities given, the values of others which are unknown, so that these latter may answer certain conditions proposed.

(9.) And these values are called Roots of the Equation.

(10.) Known quantities are usually expressed by the first letters of the alphabet, a, b, c, &c.; and unknown quantities by the last, v, x, y, &c. And this must be always understood, unless the contrary be expressed.

AXIOMS.

(11.) If equal quantities be added to equal quantities, the sums will be equal.

(12.) If equal quantities be taken from equal quantities, the remainders will be equal.

(13.) If equal quantities be multiplied by the same or equal quantities, the products will be equal.

(14.) If equal quantities be divided by the same or equal quantities, the quotients will be equal.

(15.) If the same quantity be added to and subtracted from another, the value of the latter will not be altered.

(16.) If a quantity be both multiplied and divided by another, its value will not be altered.

(17.) Any quantity may be transposed from one side of an equation to the other, by changing its sign:

Because, in this transposition, the same quantity is merely subtracted from each side of the equation; and (12) if equals be taken from equals, the remainders are equal.

Thus, if x+9=15, and 9 be subtracted from each side, x 15-9, or 6. Also, if x + b = a, and b be subtracted from each side, ra— b. And if x-c=d, and c be added to each side, r = d + c.

Also, if 5x-7=2x+2, and 2x be taken from each side, 5x-2x-7=2, or 3x-7=2; and if 7 be subtracted, or (which is the same thing) if +7 be added to each side, 3x = 2 + 7 = 9.

Also, if a + b = c − 3x, then, by subtracting -ab-3x from each side, we have r+3x=a-b+c,

COR. 1. Hence, if the signs of all the terms on each side of an equation be changed, the two sides still remain equal; because in this change every term is transposed.

COR. 2. Hence, when the known and unknown quantities are connected in an equation by the signs + or -, -, they may be separated by transposing the known quantities to one side, and the unknown to the other.

COR. 3. Hence also, if any quantity be found on both sides of an equation, it may be taken away from each; thus, if x+y=5+y, then x=5. b = c + d − b, then a = c + d.

If a

(18.) If every term on each side of an equation be multiplied by the same quantity, the results will be equal :

Because in multiplying every term on each side by any quantity, the value of the whole side is multiplied by that quantity; and (13) if equals be multiplied by the same quantity, the products will be equal.

Thus, if x=5+a, then 6x=30+6a, by multiplying every term by 6.

COR. 1. Hence an equation, of which any part is fractional, may be reduced to an equation expressed in integers, by multiplying every term by the denominator of the fraction. If there be more fractions than one in the given equation, it may be so reduced by multiplying every term either by the