Let "a" and "b" be the roots. Then, according to the condition


  a   - b   = 2      (1)

  a^2 - b^2 = 5      (2)


We can present equation (2) in the form 

    (a+b)*(a-b) = 5.


Replace (a-b) here by 2, based on equation (1), Then you get

    2*(a+b) = 5,


which implies 

    a + b = 5/2 = 2.5.


Hence, instead of (1) and (2), we have this, more simple system of equations

    a - b = 2,      (3)

    a + b = 2.5.    (4)


Add these equations. You will get  2a = 4.5,  which implies  a = 4.5/2 = 2.25.


So, one root is  a= 2.25.  Hence, the other root is 2 unit less  b= 2.25 - 2 = 0.25.


Thus the roots are  a= 2.25, b = 0.25.


Then the quadratic equation under the question is

    (x- 2.25)*(x-0.25) = 0.


You can transform it to any other equivalent form.