Question

which equation can be rewritten as $x + 4 = x^2$? assume $x > 0$.

$\bigcirc\\ \sqrt{x} + 2 = x$

$\bigcirc\\ \sqrt{x + 2} = x$

$\bigcirc\\ \sqrt{x + 4} = x$

$\bigcirc\\ \sqrt{x^2 + 16} = x$

Explanation:

Step1: Analyze Option A

Start with the equation $\sqrt{x} + 2 = x$. Isolate the square root: $\sqrt{x}=x - 2$. Square both sides: $x=(x - 2)^2=x^{2}-4x + 4$. Rearrange: $x^{2}-5x + 4 = 0$. This is not $x + 4=x^{2}$, so A is incorrect.

Step2: Analyze Option B

Start with $\sqrt{x + 2}=x$. Square both sides: $x + 2=x^{2}$. Rearrange: $x^{2}-x - 2 = 0$. Not $x + 4=x^{2}$, so B is incorrect.

Step3: Analyze Option C

Start with $\sqrt{x + 4}=x$. Square both sides: $x + 4=x^{2}$. This matches the target equation, so we can check D for completeness.

Step4: Analyze Option D

Start with $\sqrt{x^{2}+16}=x$. Square both sides: $x^{2}+16=x^{2}$. Subtract $x^{2}$: $16 = 0$, which is false. So D is incorrect.

Answer:

$\boldsymbol{\sqrt{x + 4}=x}$ (the third option, or the option with $\sqrt{x + 4}=x$)