If $x = 5 + 2\sqrt{6}$, then $\frac{(x - 1)}{\sqrt{x}}$ is equal to

### Concept & Formula The key to simplifying expressions with surds under square roots is to express the inner surd as a perfect square. This is done by splitting the integer part into two numbers that add up to it, and whose product relates to the surd part. $$(a + b)^2 = a^2 + b^2 + 2ab$$ ### Step-by-Step Solution * **Given:** $$x = 5 + 2\sqrt{6}$$ Expression to evaluate: $$\frac{x - 1}{\sqrt{x}}$$ * **Calculation / Deduction:** 1. First, simplify the given expression by separating the terms: $$\frac{x - 1}{\sqrt{x}} = \frac{x}{\sqrt{x}} - \frac{1}{\sqrt{x}} = \sqrt{x} - \frac{1}{\sqrt{x}}$$ 2. Next, find the square root of $x$. We need to express $5 + 2\sqrt{6}$ in the form of $(a + b)^2$. 3. Notice that $5 = 3 + 2$ and $\sqrt{6} = \sqrt{3 \times 2}$. $$x = 3 + 2 + 2\sqrt{3}\sqrt{2} = (\sqrt{3})^2 + (\sqrt{2})^2 + 2(\sqrt{3})(\sqrt{2})$$ $$x = (\sqrt{3} + \sqrt{2})^2$$ 4. Taking the square root of both sides: $$\sqrt{x} = \sqrt{3} + \sqrt{2}$$ 5. Now find its reciprocal, $\frac{1}{\sqrt{x}}$. Rationalize the denominator: $$\frac{1}{\sqrt{3} + \sqrt{2}} \times \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} - \sqrt{2}} = \frac{\sqrt{3} - \sqrt{2}}{3 - 2} = \sqrt{3} - \sqrt{2}$$ 6. Substitute these back into our simplified expression: $$\sqrt{x} - \frac{1}{\sqrt{x}} = (\sqrt{3} + \sqrt{2}) - (\sqrt{3} - \sqrt{2})$$ $$\sqrt{x} - \frac{1}{\sqrt{x}} = \sqrt{3} + \sqrt{2} - \sqrt{3} + \sqrt{2} = 2\sqrt{2}$$ ### Exam Strategy & Shortcut **Algebraic Manipulation:** Instead of finding $\sqrt{x}$ first, you can manipulate the final required expression. If $y = \sqrt{x} - \frac{1}{\sqrt{x}}$, then squaring both sides gives $y^2 = x + \frac{1}{x} - 2$. Given $x = 5 + 2\sqrt{6}$, the reciprocal is $\frac{1}{x} = 5 - 2\sqrt{6}$ (since $5^2 - (2\sqrt{6})^2 = 25 - 24 = 1$). So, $x + \frac{1}{x} = (5 + 2\sqrt{6}) + (5 - 2\sqrt{6}) = 10$. Then, $y^2 = 10 - 2 = 8$. Thus, $y = \sqrt{8} = 2\sqrt{2}$. This completely avoids finding the square root of a surd! ### Common Pitfall A frequent mistake is plugging the raw value of $x$ directly into the fractional expression without simplifying it first. This leads to messy rationalization of a complex denominator like $\sqrt{5 + 2\sqrt{6}}$, which consumes a lot of exam time and often leads to arithmetic errors. ### Final Answer **Therefore, the correct answer is $2\sqrt{2}$.**