Question:সরল কর:

 ক. `(x^2 + 2x - 15)/(x^2 + x - 12) -: (x^2 - 25)/(x^2 - x - 20) xx (x - 2)/(x^2 - 5x + 6)`

 খ. `(x/(x - y) - x/(x + y)) -: (y/(x - y) - y/(x + y)) + ((x + y)/(x - y) + (x - y)/(x + y)) `

                                                      `-: ((x + y)/(x - y) - (x - y)/(x + y))`

 গ. `(x^2 + 2x - 3)/(x^2 + x - 2) -: (x^2 + x - 6)/(x^2 - 4)`

 ঘ. `(a^4 - b^4)/(a^2 + b^2 - 2ab) xx ((a + b)^2 - 4ab)/(a^3 - b^3) `

                                                     `-:  (a + b)/(a^2 + ab + b^2)` 

Answer ক. `(x^2 + 2x - 15)/(x^2 + x - 12) -: (x^2 - 25)/(x^2 - x - 20) xx (x - 2)/(x^2 - 5x + 6)`

   `= (x^2 + 5x - 3x - 15)/(x^2 + 4x - 3x - 12) -: (x^2 - 5^2)/(x^2 - 5x + 4x - 20) xx (x - 2)/(x^2 - 3x - 2x + 6)`

   `= (x(x + 5) - 3(x + 5))/(x(x + 4) - 3(x + 4) - 3(x + 4)) -: ((x + 5)(x - 5))/(x(x - 5) + 4(x - 5)) `

                                                                                            `xx (x - 2)/(x(x - 3) - 2(x - 3))`

 
 `= ((x + 5)(x - 3))/((x + 4) (x - 3)) -: ((x + 5)(x - 5))/((x - 5)(x + 4)) xx (x - 2)/((x - 3) (x - 2))`

  `= (x + 5)/(x + 4) xx (x + 4)/(x + 5) xx 1/(x - 3)`

  `= 1/(x - 3)`

     উত্তর: `1/(x - 3)`

 খ. `(x/(x - y) - x/(x + y)) -: (y/(x - y) - y/(x + y)) + ((x + y)/(x - y) + (x - y)/(x + y)) `

                                                            `-: ((x + y)/(x - y) - (x - y)/(x + y))`

   `= (x(x + y) - x(x - y))/((x - y)(x + y)) -: (y(x + y) - y(x - y))/((x - y)(x + y))`

                    `+ ((x + y)^2 + (x - y)^2)/((x - y) (x + y)) -: ((x + y)^2 - (x - y)^2)/((x - y) (x + y))`

   `= (x^2 + xy - x^2 + xy)/((x - y) (x + y)) -: (xy + y^2 - xy + y^2)/((x - y)(x + y))`

          `+ (x^2 + 2xy + y^2 + x^2 - 2xy + y^2)/((x - y)(x + y)) -: (x^2 + 2xy + y^2 - x^2 + 2xy - y^2)/((x - y)(x + y))`

   `= (2xy)/((x - y)(x + y)) xx ((x - y)(x + y))/(2y^2) + (2x^2 + 2y^2)/((x - y)(x + y)) -: (4xy)/((x - y)(x + y))`

   `= x/y + (2(x^2 + y^2))/((x + y)(x - y)) xx ((x - y)(x + y))/(4xy)`

   `= x/y + (x^2 + y^2)/(2xy)`

   `= (2x^2 + x^2 + y^2)/(2xy)`

   `= (3x^2 + y^2)/(2xy)`

     উত্তর: `(3x^2 + y^2)/(2xy)`

 গ. `(x^2 + 2x - 3)/(x^2 + x - 2) -: (x^2 + x - 6)/(x^2 - 4)`

     `= (x^2 + 3x - x - 3)/(x^2 + 2x - x - 2) -: (x^2 + 3x - 2x - 6)/(x^2 - 2^2)`

     `= (x(x + 3) - 1(x + 3))/(x(x + 2) - 1(x + 2)) -: (x(x + 3) - 2(x + 3))/((x + 2)(x - 2))`

     `= ((x + 3)(x - 1))/((x + 2)(x - 1)) -: ((x + 3)(x - 2))/((x + 2)(x - 2))`

     `= (x + 3)/(x + 2) -: (x + 3)/(x + 2)`

     `= (x + 3)/(x + 2) xx (x + 2)/(x + 3)`

       = 1

         উত্তর: 1

 ঘ. `(a^4 - b^4)/(a^2 + b^2 - 2ab) xx ((a + b)^2 - 4ab)/(a^3 - b^3) -:  (a + b)/(a^2 + ab + b^2)`

    `= ((a^2 + b^2)(a^2 - b^2))/(a^2 - 2ab + b^2) xx (a^2 + 2ab + b^2 - 4ab)/((a - b)(a^2 + ab + b^2) -: (a + b)/(a^2 + ab + b^2)`

    `= ((a - b)(a + b)(a^2 + b^2))/(a - b)^2 xx (a^2 - 2ab + b^2)/((a - b)(a^2 + ab + b^2)) xx (a^2 + ab + b^2)/(a + b)`

   `= ((a - b)(a + b)(a^2 + b^2))/(a - b)^2 xx (a - b)^2/((a - b)(a^2 + ab + b^2)) xx (a^2 + ab + b^2)/(a + b)`

   `= a^2 + b^2`

     উত্তর: ` a^2 + b^2` 

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