5 Useful tricks in Algebra - Vikas Saini



  • Below are the five useful tricks that comes in extra handy.
     

    Type - 1

    If [m/(x+a) (x+b)]+[n/(x+b)(x+c)]+[o/(x+c)(x+a)] = 0.

    x = - (mc+na+ob)/(m+n+o)

    or

    x = n x vanished number / (n1 + n2 + n3)

    Here n = numerator.

    Q. 1/(x -1)(x-2) + 2/(x-2)(x-3) + 3/(x-3)(x-1)  = 0.

    By direct formula

    x = - [1x(-3)+2x(-1)+3x(-2)] / (1+2+3)

    x = - (-11)/6

    x = 11/6.

    Q. 1/(x^2+3x+2) + 5/(x^2+5x+6) + 3/(x^2+4x+3) = 0

    solution :-

    1/(x+2)(x+1) + 5/(x+2)(x+3) + 3/(x+1)(x+3)

    x = - [ 1 x 3 + 5 x 1 + 3 x 2 ] / ( 1+5+3)

    x = - 14/9.

    Type 2

    1/xy + 1/xz  = 1/xw + 1/yz

    if x,y,z & w are in Arithmetic progression.

    Then last term + 2 x second last term = 0.

    Q. 1/(x+2)(x+3) + 1/(x+2)(x+4) = 1/(x+2) (x+5) + 1/(x+3)(x+4)

    Here 2,3,4,5 in AP.

    So last term + 2 x second last term = 0.

    (x+5) + 2(x+4) = 0

    x+5+2x+8 = 0

    x = -13/3.

    Q. 1/(x+1)(x+3) + 1/(x+1)(x+5) = 1/(x+1)(x+7) + 1/(x+3)(x+5)

    solution :-

    1,3,5,7 are in AP.

    So (x+7)+2(x+5) = 0.

    x+7+2x+10 = 0.

    x = -17/3.

    Type 3 :-

    if ax^2+bx+c / dx^2+ex+f = ax+b / dx+e

    Then

    c / d = ax+b / dx+e.

    Q. 3x^2+5x+8 / 5x^2+6x+12 = 3x+5/5x+6

    solution :-

    = > 8 / 12 = 3x+5 / 5x+6

    = > 2 / 3 = 3x+5 / 5x+6

    = > 10x+12 = 9x+15

    = > x = 3.

    Q. 2-2x-3x^2 / 2-5x-6x^2 = 3x+2 / 6x+5

    solution :-

    here (3x^2+2x)+2 / -(6x^2+5x)+2 = 3x+2 / 6x+5

    = > 3x+2/6x+5 = 2/2

    = > 3x+2 = 6x+5

    = > x = -1.

    Q. (x+2)(x+3)(x+11)= (x+4)(x+5)(x+7)

    solution :-

    = > (x+2)(x+3)/(x+4)(x+7) = (x+4)/(x+11)

    = > x^2+5x+6 / x^2+11x+28 = x+4/x+11

    = > 6/28 = (x+5)/(x+11)

    = > 3 / 14 = (x+5) / (x+11)

    = > 3x+33 = 14x+70

    = > -11x = 37

    = > x = -37/11.

    Type - 4

    To find the sum of given series

    y/(x+a)(x+b) + z/(x+b)(x+c)......

    Sn = [{y+z......(n times)}/ (x+a) {x+a+n(b-a)} ]

    Q. 1/(x+3)(x+4) + 1/(x+4)(x+5) + 1/(x+5)(x+6).........

    Find S4

    Solution :-

    S4 = 1+1+1+1/ (x+3) {x+3+(4-3)4}

          = 4/(x+3)(x+7).

    Q. 1/(x^2-3x+2) + 1/(x^2-5x+6)......

    Find S5

    solution :-

    1/(x-1)(x-2) + 1/(x-2)(x-3)

    S5 = 1x5 / (x-1) {x-1+(-3+2)5}

          = 5 / (x-1)(x-6)

    Q. 1/2x3 + 1/3x4 + 1/4x5...........1/19x20

    Solution :-

    Here Terms = 19-1 = 18.

    S18 = 18 / 2 x {2+(3-2)x18}

            = 18 / 2 x 20

            = 18 / 40.

    Type 5

    (b-a)/(x+a)(x+b) + (c-b)/(x+b)(x+c) ............................+ (z-y)/(x+y)(x+z)

    Then Sum = (z-a)/(x+a)(x+z).

    Q. 1/7x8 + 2/8x10 + 14/24x10.

    Find sum.

    solution :-

    sum = 1+2+14/7x24

    = 17/168.

    Q. 3/7x10 + 9/10x19 + 27/19x46 + 99/46x145.

    Find sum.

    solution :-

    sum = 3+9+27+99/7x145

    = 138 / 1015.

    Q. 3/4 + 5/36 + 7/144 + 9/400 ....19/8100.

    solution :-

    By direct formula

    3/4x1 + 5/4x9 + 7/9x16 + 9/16x25.....................19/81 x 100

    = 3+5+........19 / 1 x 100

    = (3+19)x9/2 x 100

    = 99/100.

     


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