Question

which of the following is a radical equation?

\\(x\sqrt{3} = 13\\)
\\(x + \sqrt{3} = 13\\)
\\(\sqrt{x} + 3 = 13\\)
\\(x + 3 = \sqrt{13}\\)

Explanation:

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<reasoning>

Define a radical equation

Using the Solving Radical Equations knowledge point
An equation is classified as a radical equation if and only if it contains a variable inside a radical (such as a square root, cube root, etc.). If the radical contains only a constant number, the equation is linear or polynomial, not radical.

Analyze each option

• Option 1: \(x\sqrt{3} = 13\)

The variable \(x\) is outside the radical. The radical contains only the constant \(3\). This is a linear equation.

• Option 2: \(x + \sqrt{3} = 13\)

The variable \(x\) is outside the radical. The radical contains only the constant \(3\). This is a linear equation.

• Option 3: \(\sqrt{x} + 3 = 13\)

The variable \(x\) is inside the square root radical. Therefore, this is a radical equation.

• Option 4: \(x + 3 = \sqrt{13}\)

The variable \(x\) is outside the radical. The radical contains only the constant \(13\). This is a linear equation.
</reasoning>

<answer>
<mcq-option>(A) \(x\sqrt{3} = 13\)</mcq-option>
<mcq-option>(B) \(x + \sqrt{3} = 13\)</mcq-option>
<mcq-correct>(C) \(\sqrt{x} + 3 = 13\)</mcq-correct>
<mcq-option>(D) \(x + 3 = \sqrt{13}\)</mcq-option>
</answer>

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Answer:

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<reasoning>

Define a radical equation

Using the Solving Radical Equations knowledge point
An equation is classified as a radical equation if and only if it contains a variable inside a radical (such as a square root, cube root, etc.). If the radical contains only a constant number, the equation is linear or polynomial, not radical.

Analyze each option

• Option 1: \(x\sqrt{3} = 13\)

The variable \(x\) is outside the radical. The radical contains only the constant \(3\). This is a linear equation.

• Option 2: \(x + \sqrt{3} = 13\)

The variable \(x\) is outside the radical. The radical contains only the constant \(3\). This is a linear equation.

• Option 3: \(\sqrt{x} + 3 = 13\)

The variable \(x\) is inside the square root radical. Therefore, this is a radical equation.

• Option 4: \(x + 3 = \sqrt{13}\)

The variable \(x\) is outside the radical. The radical contains only the constant \(13\). This is a linear equation.
</reasoning>

<answer>
<mcq-option>(A) \(x\sqrt{3} = 13\)</mcq-option>
<mcq-option>(B) \(x + \sqrt{3} = 13\)</mcq-option>
<mcq-correct>(C) \(\sqrt{x} + 3 = 13\)</mcq-correct>
<mcq-option>(D) \(x + 3 = \sqrt{13}\)</mcq-option>
</answer>

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