Question

which equation can be rewritten as ( x + 4 = x^2 )? assume ( x > 0 ).
options:
( sqrt{x + 2} = x )
( sqrt{x + 4} = x )
( sqrt{x^2 + 16} = x )
( sqrt{x + 2} = x ) (note: possible typo in original, but ocr captures as is)

Explanation:

Step1: Analyze each option

We need to square both sides of each square - root equation (since \(x>0\), squaring both sides is a valid operation that will not introduce extraneous solutions in terms of the domain we are considering) and see which one gives \(x + 4=x^{2}\).

Option 1: \(\sqrt{x + 2}=x\)

Square both sides: \((\sqrt{x + 2})^{2}=x^{2}\), which simplifies to \(x + 2=x^{2}\). This is not the equation we want.

Option 2: \(\sqrt{x + 4}=x\)

Square both sides: \((\sqrt{x + 4})^{2}=x^{2}\), which simplifies to \(x+4 = x^{2}\). This is the equation we are looking for.

Option 3: \(\sqrt{x^{2}+16}=x\)

Square both sides: \((\sqrt{x^{2}+16})^{2}=x^{2}\), which simplifies to \(x^{2}+16=x^{2}\). Subtract \(x^{2}\) from both sides, we get \(16 = 0\), which is a contradiction.

Option 4: (There is a repetition of the first option, \(\sqrt{x + 2}=x\) which we already analyzed and it gives \(x + 2=x^{2}\))

Answer:

The equation \(\boldsymbol{\sqrt{x + 4}=x}\) (the middle - middle option, or the option with the square - root of \(x + 4\) equal to \(x\)) can be rewritten as \(x + 4=x^{2}\) for \(x>0\).