Igor <igor at iuav.it> writes:
> Dear,
> Who can help me?
> I need a step by step solution of the following equation:
> sqrt(3-sqrt(x-1))-sqrt(x-sqrt(2)).
>
> The only result I get is the following: sqrt(x-sqrt(2))=sqrt(3-sqrt(x-1)).
>
> Now, how can i get a right result for x by a step by step solution?
> Many many thanks.
> Igor
Dear Igor,
You haven't explained why you need a step-by-step solution of this
problem, which I suspect makes most readers wonder whether you're doing
homework. I'm one such reader, so I'll only give slightly vague hints:
Firstly, notice that
sqrt(A) = sqrt(B)
implies that A = B. Of course, the converse isn't true (think about
this: what about positive and negative solutions?)
So you need to find all the solutions to
x - sqrt(2) = 3 - sqrt(x-1)
It might not look like it, but this is a quadratic equation. Just asking
Maxima to solve it won't get you anything helpful, but try squaring it
out:
(%i1) x - sqrt(2) = 3 - sqrt(x-1);
(%o1) x - sqrt(2) = 3 - sqrt(x - 1)
(%i2) solve(%, sqrt(x-1));
(%o2) [sqrt(x - 1) = - x + sqrt(2) + 3]
(%i3) first(%)^2;
2
(%o3) x - 1 = (- x + sqrt(2) + 3)
You should be able to expand and solve this quadratic by hand, or Maxima
can do it for you. Don't forget that you now have to work back down the
pair of square roots: Since it's a quadratic, you've ended up with up to
two solutions. These may or may not be solutions to the original
problem.
Yours,
Rupert
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