Question

Solve for X. Check for extraneous answers.
sqrt{x + sqrt{2x}}=sqrt{2}

Answer 1

square both sides get
\[x+\sqrt{2x}=2\] subtract \(x\) from both sides and square again
\[\sqrt{2x}=2-x\]
\[2x=(2-x)^2=4-4x+x^2\]

Answer 2

then set equal to zero and solve the quadratic
\[x^2-4x+4=2x\]
\[x^2-6x+4=0\]

Answer 3

would x= 2?

Answer 4

oh no you have to either use the quadratic formula or complete the square

Answer 5

and you also have to check your answer
\[x^2-6x=-4\]
\[(x-3)^2=-4+9=5\]
\[x-3=\pm\sqrt5]\]
\[x=3\pm\sqrt5\]

Answer 6

now you have to check them
it will turn out that \(3-\sqrt5\) works, while \(3+\sqrt5\) does not

Answer 7

Thank you very much! Without you, I would have been stuck on this question for the whole night! Thank you!