How to determine # of solutions to sqroot of x+5=x?

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SUMMARY

The equation √x + 5 = x has only one real solution, despite the quadratic equation x^2 - x - 5 = 0 yielding two solutions. The discrepancy arises because squaring both sides introduces extraneous solutions that do not satisfy the original equation. To verify solutions, one must substitute back into the original equation. Graphing the functions can also provide a visual confirmation of the number of solutions.

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Homework Statement



How many real solutions does the equation √x+5=x have?

Homework Equations



n/a

The Attempt at a Solution



I squared both sides so I got x+5=x^2 then I formed a quadratic equation: x^2-x-5=0

I got two real solutions but the answer is one.

I know it has something to do with x+5 being sqrooted but I don't know how to go about solving it.
 
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Well you did it correctly, but when you squared it, you picked up an additional answer which satisfies the quadratic formed. But really you want your value to satisfy the original equation. So you will need to sub back in the values to see if you get the same number on the left hand side and right hand side of the equations.

It's like you start off with x= 1 and then get x^2-1=0 = (x+1)(x-1) such that x= 1 or -1.

Alternatively, you could have sketched/graphed the lines.
 

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