composed; and the parts that stand on the right and left of the sign, are called the two members, or sides, of the equation. Thus, if x = a + b, the terms are x, a, and b; and the meaning of the expression is, that some quantity x, standing on the lefthand side of the equation, is equal to the sum of the quantities a and b on the righthand side. A simple equation is that which contains only the first power of the unknown quantity: as, x + α = 3b, or ax = bc, or 2x + 3a2: 5b2; = Where a denotes the unknown quantity, and the other letters, or numbers, the known quantities. A compound equation is that which contains two or more different powers of the unknown quantity: as, x2 + ax = b, or x3- 4x2+3x= 25. Equations are also divided into different orders, or receive particular names, according to the highest power of the unknown quantity contained in any one of their terms: as quadratic equations, cubic equations, biquadratic equations, &c. Thus, a quadratic equation is that in which the unknown quantity is of two dimensions, or which rises to the second power: as, x2 = 20; x2 + ax = b, or 3x2+10x = 100. = = C. A cubic equation is that in which the unknown quantity is of three dimensions, or which rises to the third power: as, x3 27; 2x3 3x=35; or x3 ax2 + bx: A biquadratic equation is that in which the unknown quantity is of four dimensions, or which rises to the fourth power: as x 25; 5x1 — 4x=6; or x1 — ax3 + bx2 - cx=d. And so on for equations of the 5th, 6th, and other higher orders, which are all denominated according to the highest power of the unknown quantity contained in any one of their terms. The root of an equation is such a number or quantity, as, being substituted for the unknown quantity, will make both sides of the equation vanish, or become equal to each other. A simple equation can have only one root; but every compound equation has as many roots as it contains dimensions, or as is denoted by the index of the highest power of the unknown quantity, in that equation. 15, the root, or Thus, in the quadratic equation x2+2x= value of x, is either + 3 or 5; and, in the cubic equation 24, the roots are 2, 3, and 4, as will be - 9x2+26x found by substituting each of these numbers for x. In an equation of an odd number of dimensions, one of its : roots will always be real whereas, in an equation of an even number of dimensions, all its roots may be imaginary; as roots of this kind always enter into an equation by pairs. Such are the equations x2 9x2+10x+ 50 = 0.* OF THE RESOLUTION OF SIMPLE EQUATIONS, Containing only one unknown Quantity. THE resolution of simple, as well as of other equations, is the disengaging the unknown quantity, in all such expressions, from the other quantities with which it is connected, and making it stand alone, on one side of the equation, so as to be equal to such as are known on the other side; for the performing of which, several axioms and processes are required, the most useful and necessary of which are the following:-† CASE I. Any quantity may be transposed from one side of an equation to the other, by changing its sign; and the two members, or sides, will still be equal. Thus, if x + 3 = 7; then will x = my. 3, or x = 4. To the properties of equations abovementioned, we may here farther add: 1. That the sum of all the roots of any equation is equal to the coefficient of the second term of that equation, with the sign changed. 2. The sum of the products of every two of the roots, is equal to the coefficient of the third term, without any change in its sign. 3. The sum of the products of every three terms of the roots, is equal to the coefficient of the fourth term, with its sign changed. 4. And so on, to the last, or absolute term, which is equal to the product of all the roots, with the sign changed or not, according as the equation is of an odd or an even number of dimensions. See, for a more particular account of the general theory of equations, Vol. II. of Bonnycastle's Treatise on Algebra, 8vo., 1820; or Ryan's Elementary Treatise on Algebra, 12mo., 1824.-ED. The operations required for the purpose here mentioned, are chiefly such as are derived from the following simple and evident principles:1. If the same quantity be added to, or subtracted from, each of two equal quantities, the results will still be equal; which is the same, in effect, as taking any quantity from one side of an equation, and placing it on the other side, with a contrary sign. 2. If all the terms of any two equal quantities, be multiplied or divided, by the same quantity, the products, or quotients thence arising, will be equal. 3. If two quantities, either simple or compound, be equal to each other, any like powers, or roots, of them will also be equal. All of which axioms will be found sufficiently illustrated by the processes arising out of the several examples annexed to the six different cases given in the text. 4+6 = = 8; then will x = 8+4 66. And, if x 8= 3x20; then 4x - d. 3x =208, and From this rule it also follows, that if a quantity be found on each side of an equation, with the same sign, it may be left out of both of them; and that the signs of all of the terms of any equation may be changed from + to +, without altering its value. or from to = 7. Thus, if x + 5 = 7+5; then, by cancelling, x = And if a x = b − c; then, by changing the signs, b, or x = a + c − b. 5. Given 7x+83=6x+4 to find x. CASE II. Ans. x=- - 1. If the unknown quantity, in any equation, be multiplied by any number, or quantity, the multiplier may be taken away, by dividing all the rest of the terms by it; and if it be divided by any number, the divisor may be taken away, by multiplying all the other terms by it. CASE III. Any equation may be cleared of fractions, by multiplying each of its terms, successively, by the denominators of those fractions, or by multiplying both sides by the product of all the denominators, or by any quantity that is a multiple of them. х Thus, if += 5, then, multiplying by 3, we have x + 3 4 х 3x 4 = 15; and this, multiplied by 4, gives 4x+3x= 60; whence, by addition, 7x = 60, or x = х And, if + = 10; then, multiplying by 12, (which is a 4 6 multiple of 4 and 6,) 3x + 2x = 120, or 5x It also appears, from this rule, that if the same number, or quantity, be found in each of the terms of an equation, either as a multiplier or divisor, it may be expunged from all of them, without altering the result. Thus, if ax = = ab + ac; then by cancelling, x = b + e. х b And if + = 3x 2 3 + + х 3 + = 20 2 3 x+1 x+2 4. Given + to find x. 2 3 4 If the unknown quantity, in any equation, be in the form of a surd, transpose the terms so that this may stand alone, on one side of the equation, and the remaining terms on the other (by Case I.); then involve each of the sides to such a power as corresponds with the index of the surd, and the equation will be rendered free from any irrational expression. Thus, if x-2= 3; then will √ x= 3+25, or, by squaring, x5 = 25. And if √(3x + 4) = 5; then will 3x + 4 = 25, or 3x=25 Also, if (2x+3)+48; then will 3 (2x+3)=8-4 = 4, or 2x+3=43=64, and consequently 2x=64-3=61, If that side of the equation which contains the unknown quantity, be a complete power, the equation may be reduced. to a lower dimension, by extracting the root of the said power on both sides of the equation. 9; and if x3 = 27, then 33 9 24; then 3x2= 24 +9=33, or x3 : 3 = 11, and consequently x = √11. And, if x2+6x+9=27; then, since the lefthand side of the equation is a complete square, we shall have, by extracting the roots, + 3 = √27 : √ (9 × 3) = 3√3, or x = 3√3 _3. = EXAMPLES FOR PRACTICE. 1. Given 9x36 30 to find x. 2. Given a3 +936 to find x. 3. Given x2 + x + 1/ Ans. x 2. Ans. x=3 4. Given x2 + ax + b2 to find x. Ans. 4 5. Given x2+14x + 49 121 to find x. Ans. x 4. 8124 |