√(x²) = |x| — Radical Distance Trick
October 28, 2024 2025-10-29 15:27√(x²) = |x| — Radical Distance Trick
Concept: \( \sqrt{x^{2}} = |x| \) — Turn Radicals into Distance
CAT • XAT • ~5–7 min
Algebra
Radicals
Distance Trick
📘 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.
When you see \( \sqrt{x+a-b\sqrt{x-c}} \), try \( t=\sqrt{x-c}\Rightarrow x=t^{2}+c \). You will often get \( |t-p|+|t-q|=k \).
🧩 Worked Example (CAT style)
Nature of solutions of
\[ \sqrt{x+3-4\sqrt{x-1}}+\sqrt{x+8-6\sqrt{x-1}}=1. \]
- Let \( t=\sqrt{x-1} \) (\(t\ge0\)) so \( x=t^{2}+1 \).
- Then \[ \sqrt{t^{2}+4-4t}+\sqrt{t^{2}+9-6t} = \sqrt{(t-2)^{2}}+\sqrt{(t-3)^{2}} = |t-2|+|t-3|=1. \]
- On a number line, \( |t-2|+|t-3|=1 \iff 2\le t\le3 \).
- 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
Tip: Convert each radical into an absolute value using \( \sqrt{(t-a)^2}=|t-a| \). Decide intervals where the absolute values open with \(\pm\) signs or use the distance sum property directly.
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