Question

If \( \sqrt{3}x - 2 = 2\sqrt{3} + 4 \), then the value of \( x \) is:

• A. \( 1 + \sqrt{3} \)
• B. \( 2(1 + \sqrt{3}) \)
• C. \( 1 - \sqrt{3} \)
• D. \( 2(1 - \sqrt{3}) \)

Hint:

When solving equations with square roots, isolate the term with the square root and simplify carefully. Rationalize the denominator if necessary.

Answer 1

The Correct Option is B

We are given the equation: \[ \sqrt{3}x - 2 = 2\sqrt{3} + 4 \] First, isolate the term with \( x \) on one side by adding 2 to both sides: \[ \sqrt{3}x = 2\sqrt{3} + 6 \] Now, divide both sides by \( \sqrt{3} \): \[ x = \frac{2\sqrt{3} + 6}{\sqrt{3}} = \frac{2\sqrt{3}}{\sqrt{3}} + \frac{6}{\sqrt{3}} = 2 + \frac{6}{\sqrt{3}} \] Next, simplify \( \frac{6}{\sqrt{3}} \). Multiply numerator and denominator by \( \sqrt{3} \): \[ \frac{6}{\sqrt{3}} = \frac{6\sqrt{3}}{3} = 2\sqrt{3} \] Thus, we have: \[ x = 2 + 2\sqrt{3} \] Factor out a 2: \[ x = 2(1 + \sqrt{3}) \] Therefore, the correct answer is \( 2(1 + \sqrt{3}) \).