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my best girl melody marks - Decision Point

📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 Thus, the value of $ b $ is $\boxed{8}$.

📌 Question: A science educator designs a learning module where a student's engagement score $ E $ is modeled by $ E = \frac{3x + 4}{\sqrt{x} - 2} $. To simplify analysis, the denominator must be rationalized. What is the rationalized form of $ E $?

📌 Solution: To rationalize the denominator of $ \frac{3x + 4}{\sqrt{x} - 2} $, multiply numerator and denominator by the conjugate $ \sqrt{x} + 2 $:

📌 \frac{3x + 4}{\sqrt{x} - 2} \cdot \frac{\sqrt{x} + 2}{\sqrt{x} + 2} = \frac{(3x + 4)(\sqrt{x} + 2)}{(\sqrt{x} - 2)(\sqrt{x} + 2)}

📌 = \frac{(3x + 4)(\sqrt{x} + 2)}{x - 4}

📌 Thus, the rationalized form is:

📌 \frac{3x\sqrt{x} + 6x + 4\sqrt{x} + 8}{x - 4}

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