Algebra maths Olympiad question

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0:00 introduction
02:33 laws of surds and indices
06:56 how to expand

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[Mathicalminds]

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[kittu]

Your videos are very informative. Please create more of these types of videos, will be supporting!

[adeltorjmen]

Maybe a simpler solution :
Assume x-1 = u^2 and x-2 = v^2
We have u+v=1 => x=u^2+1 and u^2-v^2=1 and u+v=1 => x=u^2+1 and (u-v)(u+v)=1 and u+v=1 => x=u^2+1 and u-v=1 and u+v=1 => u=1 and v=0 and x=2

[evankassof]

Another possible approach: the difference of two squares is never 1 except when it is 1 and 0. That forces one of the terms to zero, so you can try x=1 => sqrt(1-1) + sqrt(1-2) =/= 1. Then try making the other term zero by setting x=2 => sqrt(2-1) + sqrt(2-2) = sqrt(1)+sqrt(0) = 1. The approach in the video is very thorough though/general which is good!

Edit: There's an error in the way I explained the above:

The SUM of two square-root is never 1 except when it is 1 and 0. That forces one of the terms to be zero because a square-root can't be negative. Therefore, the rest of the above makes sense lol

[hpcuthulu]

square both sides of the equation
you end up with: x-1 + x-2 = 1
2x-3=1
2x=4
x=2