What is the solution for x? sqrt(x+5)=x^2-5

The solution is very long a little tough...but I will summarize...

square both sides to get:

Simplify and rewrite as

We wish to write this as a product of square polynomials, so after some careful thought and experimentation we use



When we expand this version and compare coefficients we can determine that

1) 
2) 
3) 

This nonlinear can be solved by substitution. It has a rational solution of




therefore we can write:


and that means


so that


and we know there are only two solutions so...

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