Determine whether there exists a real number x satisfying

sqrt(x-2) = 3 - 2*sqrt(x).

If there is, determine this number.

So I square both sides which gives

x-2 = 9 - 12*sqrt(x) + 4x

= 12*sqrt(x) = 11 + 3x

Squaring both sides gives

x = (121+66x+9x^2)/12

which clearly doesn't have any real solutions, as the book's answer gives. However, I entered the original equation into WolframAlpha and it returned a real solution: 13/3 - 4/sqrt(3)

Who's wrong?

Question

Determine whether there exists a real number x satisfying

sqrt(x-2) = 3 - 2*sqrt(x).

If there is, determine this number.

So I square both sides which gives

x-2 = 9 - 12*sqrt(x) + 4x

=> 12*sqrt(x) = 11 + 3x

Squaring both sides gives

x = (121+66x+9x^2)/12

which clearly doesn't have any real solutions, as the book's answer gives. However, I entered the original equation into WolframAlpha and it returned a real solution: 13/3 - 4/sqrt(3)

Who's wrong?

anonymous · Answer

sqrt(x-2) = 3 - 2*sqrt(x).
square both sides which gives 
x-2 =9 - 12*sqrt(x) + 4x
12*sqrt(x) = 11 + 3x
square both side gives
x = (121+66x+9x^2)/144

anonymous · Answer

got your mistake?

anonymous · Answer

Ah! I just caught my error! Yes thank you I just realized I didn't square the denominator in the last bit.

anonymous · Answer

Welcome