Question

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

• \\(\sqrt{x} + 2 = x\\)
• \\(\sqrt{x + 2} = x\\)
• \\(\sqrt{x + 4} = x\\)
• \\(\sqrt{x^2 + 16} = x\\)

Explanation:

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<reasoning>

Analyze the target equation

Using the Solving Radical Equations and Radical Equation Definition knowledge points
\[

$$\begin{aligned} &x + 4 = x^2 \quad (\text{for } x > 0) \end{aligned}$$

\]

Test the given options by squaring both sides

Using the Solving Radical Equations knowledge point
\[

$$\begin{aligned} &\text{Option 1: } \sqrt{x} + 2 = x \implies \sqrt{x} = x - 2 \implies x = (x-2)^2 = x^2 - 4x + 4\\ &\text{Option 2: } \sqrt{x+2} = x \implies x + 2 = x^2\\ &\text{Option 3: } \sqrt{x+4} = x \implies x + 4 = x^2\\ &\text{Option 4: } \sqrt{x^2+16} = x \implies x^2 + 16 = x^2 \implies 16 = 0 \end{aligned}$$

\]

Identify the matching equation

Using the Solving Radical Equations knowledge point
\[

$$\begin{aligned} &\sqrt{x+4} = x \implies x + 4 = x^2 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(\sqrt{x} + 2 = x\)</mcq-option>
<mcq-option>(B) \(\sqrt{x+2} = x\)</mcq-option>
<mcq-correct>(C) \(\sqrt{x+4} = x\)</mcq-correct>
<mcq-option>(D) \(\sqrt{x^2+16} = x\)</mcq-option>
</answer>

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Answer:

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<reasoning>

Analyze the target equation

Using the Solving Radical Equations and Radical Equation Definition knowledge points
\[

$$\begin{aligned} &x + 4 = x^2 \quad (\text{for } x > 0) \end{aligned}$$

\]

Test the given options by squaring both sides

Using the Solving Radical Equations knowledge point
\[

$$\begin{aligned} &\text{Option 1: } \sqrt{x} + 2 = x \implies \sqrt{x} = x - 2 \implies x = (x-2)^2 = x^2 - 4x + 4\\ &\text{Option 2: } \sqrt{x+2} = x \implies x + 2 = x^2\\ &\text{Option 3: } \sqrt{x+4} = x \implies x + 4 = x^2\\ &\text{Option 4: } \sqrt{x^2+16} = x \implies x^2 + 16 = x^2 \implies 16 = 0 \end{aligned}$$

\]

Identify the matching equation

Using the Solving Radical Equations knowledge point
\[

$$\begin{aligned} &\sqrt{x+4} = x \implies x + 4 = x^2 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(\sqrt{x} + 2 = x\)</mcq-option>
<mcq-option>(B) \(\sqrt{x+2} = x\)</mcq-option>
<mcq-correct>(C) \(\sqrt{x+4} = x\)</mcq-correct>
<mcq-option>(D) \(\sqrt{x^2+16} = x\)</mcq-option>
</answer>

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