📘 Core Idea

The principal square root is non-negative, so \( \sqrt{x^{2}} = |x| \) (not \(x\)). This lets you turn expressions like \( \sqrt{(t-2)^{2}} \) into \( |t-2| \) and convert nested radicals into distance sums that are easy to solve.

🧩 Worked Example (CAT style)

Nature of solutions of

\[ \sqrt{x+3-4\sqrt{x-1}}+\sqrt{x+8-6\sqrt{x-1}}=1. \]

1. Let \( t=\sqrt{x-1} \) (\(t\ge0\)) so \( x=t^{2}+1 \).
2. Then \[ \sqrt{t^{2}+4-4t}+\sqrt{t^{2}+9-6t} = \sqrt{(t-2)^{2}}+\sqrt{(t-3)^{2}} = |t-2|+|t-3|=1. \]
3. On a number line, \( |t-2|+|t-3|=1 \iff 2\le t\le3 \).
4. Hence \( x=t^{2}+1 \Rightarrow 5\le x\le10 \).

Conclusion: Infinitely many real solutions (i.e., more than two).

⚠️ Pitfalls to Avoid

• Writing \( \sqrt{x^2}=x \) instead of \( |x| \).
• Ignoring the domain from \( t=\sqrt{x-1} \) (\( t\ge0 \)).
• Squaring too early; first convert to absolute values, then use number-line logic.

📝 Quick Quiz