A cubic equation is that in which the unknown quan. tity is of three dimensions, or which rises to the third power : as, 23=-27 ; 2x3 - 3x=35; or x3 - ax+bx=0. A biquadratic equation is that in which the unknown quantity is of four dimensions, or which rises to the fourth power : as, x4=25 ; 5x4 - 4x=6; or x4-ax3 + b.x2 - 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 quan. tity, 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. Thus, in the quadratic equation x? +2r=15, the root, or value of x, is either + 3 or 5 ; and, in the cubic equation x3 – 9x2 +26x=24, the roots are 2, 3, and 4, as will be found by substituting each of these numbers for 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 equa. tion by pairs. Such are the equations x2 - 6x+14=1, and x4--2203 - 3x3 +10+50=0. (2) (2) To the properties of equations above-mentioned, 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 its sign changed. 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 whicb, several axioms and processes are required, the most useful and necessary of which are the following : (a) 2. The sum of the products of every two of the røcts, is equal to the coefficient of the third term, without any change in its sign, 3. "The sum of the products of every three of the roots, is equal to the coefficient of the fourth term, with its sign changed. 4. And so un, to the last, or absolute term, which is equal to the product of all the roots, with the sign changed or not accorr ing as the equation is of an odd or an even number of dimension See, for a more particular account of the general theory equations, Vol. II. of my Treatise on Algebra, 8vo. 1813. (a) The operations required, for the purpose here mentione are chiefly such as are derived from the following simple and e dent principles : 1. If the same quantity be added to, or subtracted from, ea of two equal qnantities, the results will still be equal; which the same, in effect, as taking any quantity from one side of equation, and placing it on the other side, with a contrary sign 2. If all the terms of any two equal quantities, be multiplie or divided, by the same quantity, the products, or quotient 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 axjoms will be found sufficiently illustrated, by the processes arising out of the several examples annexed to the six different cases given in the text. CASE I. Any quantity may be transposed from one side of as equation to the other, by changing its sigo ; and the two members, or sides, will still be equal. Thus, if x+3=7; then will x=7-3, or x=4. And, if 43-8=3x+20 ; then 4x - 3x=20+8, and consequently x=28. 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 the terms of any equation may be changed from + to -, or from - to t, without altering its value. Thus, if x+5=7+5; then, by cancelling, x=7. And, if a--x=b-c; then, by changing the signs, *-=-b, or xmatc-b. CASE II. a 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. Thus, if ax=3ab - c; then will x=36 or x=8-2=6. 2 And, if -24; then 2.0-6=12, or, by division, 3 1-3-6, or 9. 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. Thus, if + i=5, then, multiplying by 3, we have x+ 3x =15; and this, multiplied by 4, gives 4x+3x=60; 60 whence, by addition, 7=60, or x= 4 4 8 7 And, if + = 10; then, multiplying by 12, (which 4 6 is a multiple of 4 and 6,) 3x+2x=120, or 5x=120, or 120 =24. 5 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+c. b And, if ; then x+b=c, or x=(-6. с CASE IV. 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 1); 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 vx=3+2=5, or, by squaring, x=52=25. And if 3x+4=5; then will 3x+4=25, or 3x= 21 25-4=21, or x= 7. 3 Also, if 72x+3+4=8; then will 72x+3=8—4 =4, or 2x+3=43=64, and consequently 2x=64-3 61 1 =61, or x= =30 CASE V. 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. Thus, if x=81; then x=81=9; and if x=27, then x=27=3. Also, if 3x-9=24 ; then 3x2=24+9=33, or 22 33 =11, and consequently x=v 11. 3 And, if x3 +6x+0=27; then, since the left hand side of the equation is a complete square, we shall have, by extracting the roots, x+3=27=V9X3=3,73, or x=33-3. CASE VI. Any analogy, or proportion, may be converted into an equation, by making the product of the two extreme terms equal to that of the two means. Thus, if 3x : 16 ::: 5 : 6; then 3x X6=16 X-5, or 80 40 4 18x=80, or 18 9 |