Solve for x: sqrt(x^2+x-6)-x+2 = sqrt(x^2-7x+10)  x is a rational no.. Plz answer fast.

Hello Ayush, x^2 + x -6 = (x+3)(x-2) and x^2 - 7x + 10 = (x-2)(x-5)
Hence problem is 
{(x+3)(x-2)}1/2 -(x-2) = {(x-2)(x-5)}1/2 
Taking (x -2)1/2 as common factor and cancelling on both sides we have
(x+3)1/2 - (x-2)1/2 = (x-5)1/2 
Squaring on both sides, (x+3) +(x+2) - 2 ​{(x+3)(x-2)}1/2 = x - 5
2x + 5 - 2​{(x+3)(x-2)}1/2  = x - 5
x + 10 = 2​{(x+3)(x-2)}1/2 
Squaring x^2 + 20x + 100 = 4 (x^2 + x -6)
Or 3x^2 - 16 x - 124 = 0
The descriminant is not a perfect square. So as x has to be a rational number, the common factor taken out and canclled out namely (x -2)1/2 = 0
===> x = 2
 
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