1 2. Required the square root of a*x* in an infinite series. 3. Convert : 1+1 into an infinite series. Ans. 1++ &c I I I 3 4. Let √xx be converted into an infinite series. x2 the first term of the root, gives - for the second term of the 22 root, which, added to 2a, gives 2a+- for the first com x2 24 x2 pound divisor, which being multiplied by -, and the product x 2a 2 taken from the first remainder x2, there remains this remainder, divided by aa, gives 4 8a3 the root, which must be added to the double of a+ -, the two first terms of the root, for the next compound divisor. And by proceeding thus, the series may be continued as far as is desired. NOTE. In order to have a true series, the greatest term of the proposed surd must be always placed first. SIMPLE EQUATIONS. AN EQUATION is when two equal quantities, differently expressed, are compared together by means of the sign placed between them. Thus, 12-5-7 is an equation, expressing the equality of the quantities 12-5 and 7. A simple equation is that, which contains only one un known quantity, in its simple form, or not raised to any power. Thus, x-a+b=c is a simple equation, containing only the unknown quantity x. Reduction of equations is the method of finding the value of the unknown quantity. It consists in ordering the equation so, that the unknown quantity may stand alone on one side of the equation without a coefficient, and all the rest, or the known quantities, on the other side. RULE 1.* Any quantity may be transposed from one side of the equation to the other, by changing its sign. Thus, if x+3=7, then will x=7-3-4 And, if x-4+6=8, then will x=8+4-6=6. Also, if x-a+b=c-d, then will x=c-d+a-b. And, in like manner, if 4x-8=3x+20, then will 4 -3x=20+8, or x=28. RULE * These are founded on the general principle of performing equal operations on equal quantities, when it is evident, that the results must still be equal; whether by equal additions, or subs tractions, or multiplications, or divisions, or roots, or powers. RULE 2. f If the unknown term be multiplied by any quantity, that quantity may be taken away by dividing all the other terms of the equation by it. Thus, if ax ab-a, then will b-1. : And if 2x+416, then will x+2=8, and x=8-2 6. In like manner, if ax+2ba3*, then will x+2b= If the unknown term be divided by any quantity, that quantity may be taken away by multiplying all the other terms of the equation by it. Thus, if=5+3, then will x=10+6=19. 2 And, if-bo-d, then will x=ab+ac-ad. a In like manner, if --2=6+4, then will 2x-6 2x 3. 18+12, and 2x=18+12+6=36, or x RULE 4. The unknown quantity in any equation may be made free from surds by transposing the rest of the terms according to the rule, and then involving each side to such a power, as is denoted by the index of the said surd. Thus, if ✓x-2=6, then will √ x=6+2=8, and x=8*=64. And, if 4x+16=12, then will 4x+16=144, and 128 4x-144-16-128, or x=-=32. 4 In 1 In like manner, if 2x+3+4=8, then will √ 2x+3 =8-4=4, And 2x+3-43-54, and 2x=64-3=61, or x =30% RULE 5. 6 If that side of the equation, which contains the unknown quantity, be a complete power, it may be reduced by extracting the root of the said power from both sides of the equation.. Thus, if x2+6x+9=25, then will x+3=√25=5, or 5-3-2. 2 And, if 3x2-9=21+3, then will 3x2 = 21+3+9= 33, and x == 11, or x=VIL 2x2 In like manner, if -+10=20, then will 2x2+30 3 2 =60, and x2+15=30, 15-30, or x*=30-15= 15, or x=15 RULE 6. Any analogy, or proportion, may be converted into an equation, by making the product of the two mean terms equal to that of the two extremes. Thus, if 3x : 16 :: 5 : 10, then will 3××10=16×5, and 30x=80, or ==2. In like manner, if 12-:-:: 4 : 1, then will 12 2 K= 4x = 2x, and 2x+x=12, or x==4 2 RULE RULE 7. If any quantity be found on both sides of the equation with the same sign, it may be taken away from them both; and if every term in an equation be multiplied or divided by the same quantity, it may be struck out of them all. b Thus, if 4x+a=b+a, then will 4x=b, and x=-. 4 And, if zax+5ab8ac, then will 3x+5b8c, and 1. Given 5x-15=2x+6; to find the value of *. First, 5x-2x=6+15 Then 3x=21 Andx==7. 2. Given 40-6-16-120-14%; to find x. First, 14x-6x=120-40+16 Then 8x 96 96 And, therefore, x=12 3. Let 5ax-36=2dx+c be given ; to find x. First, 5ax-2dx=c+3b Or 5a-2d Xx=c+3b |