Solution: To rationalize the denominator of $ \frac{3x + 4}{\sqrt{x} - 2} $, multiply numerator and denominator by the conjugate $ \sqrt{x} + 2 $:
["How to Rationalize the Denominator of $ \frac{3x + 4}{\sqrt{x} - 2} $: A Step-by-Step Guide Using the Conjugate Method", "When solving algebraic expressions involving square roots in the denominator, rationalizing the denominator is a crucial step that makes expressions simpler, cleaner, and often easier to work with in further calculations. One effective and commonly used method is to multiply both the numerator and denominator by the conjugate of the denominator. This technique is especially valuable when simplifying rational expressions in algebra and calculus.", "### The Problem: Rationalizing $ \frac{3x + 4}{\sqrt{x} - 2} $", "Consider the expression:", "$$\n\frac{3x + 4}{\sqrt{x} - 2}\n$$", "The denominator, $ \sqrt{x} - 2 $, contains a square root, which can complicate algebraic operations. Rationalizing removes the radical from the denominator, transforming it into a whole number or a simpler expression without square roots.", "---", "### The Solution: Multiply by the Conjugate", "The conjugate of a binomial expression $ a - b $ is $ a + b $. In this case, the conjugate of $ \sqrt{x} - 2 $ is $ \sqrt{x} + 2 $. Multiplying the numerator and denominator by this conjugate helps eliminate the square root in the denominator through the difference of squares formula:", "$$\n(a - b)(a + b) = a^2 - b^2\n$$", "Let’s apply this method step-by-step.", "#### Step 1: Multiply numerator and denominator by the conjugate", "$$\n\frac{3x + 4}{\sqrt{x} - 2} \cdot \frac{\sqrt{x} + 2}{\sqrt{x} + 2}\n$$", "#### Step 2: Simplify the denominator", "Use the difference of squares:", "$$\n(\sqrt{x} - 2)(\sqrt{x} + 2) = (\sqrt{x})^2 - (2)^2 = x - 4\n$$", "So the denominator becomes $ x - 4 $.", "#### Step 3: Expand the numerator", "Now distribute in the numerator:", "$$\n(3x + 4)(\sqrt{x} + 2) = 3x \cdot \sqrt{x} + 3x \cdot 2 + 4 \cdot \sqrt{x} + 4 \cdot 2\n$$", "$$\n= 3x\sqrt{x} + 6x + 4\sqrt{x} + 8\n$$", "---", "### Final Rationalized Expression", "Putting it all together:", "$$\n\frac{3x\sqrt{x} + 6x + 4\sqrt{x} + 8}{x - 4}\n$$", "This expression now has a rationalized denominator and is more manageable for further algebraic manipulation, graphing, or calculus operations.", "---", "### Why Rationalizing Denominators Matters", "- Improves readability and formality in mathematical writing.\n- Prepares expressions for integration or differentiation in calculus.\n- Facilitates algebraic simplification in complex equations.\n- Enhances accuracy in solving equations with radicals.", "---", "### Conclusion", "Rationalizing the denominator of $ \frac{3x + 4}{\sqrt{x} - 2} $ by multiplying numerator and denominator by $ \sqrt{x} + 2 $ is a clear, efficient approach using the conjugate method. This technique not only solves the immediate task but also strengthens your algebraic foundation for more advanced mathematics. Whether you're a student tackling algebra, a teacher explaining rationalization, or a self-learner mastering radicals, remembering this step will make working with square roots in denominators much easier and more intuitive.", "Try it now — rationalization is not just a formula, it’s a problem-solving tool!"]
