Does √(x²) always equal x?
Short answer: The square-root symbol gives the non-negative root, so √(x²) = |x|.
The trap
The wrong idea
“√(x²) always equals x.”
It sounds reasonable, and a lot of JEE aspirants carry it into the exam. Here's a one-question check, then the explanation that makes it click.
Quick check
Try it before you peek
√((-3)²) equals:
- -3
- 3
- ±3
Show the answer
B. 3
The square-root symbol gives the non-negative root, so √(x²) = |x|.
The click
The symbol √ means the principal, non-negative square root. For any real number y ≥ 0, √y is the one number that is at least 0 and whose square is y. So √9 = 3, never −3.
Then √((−3)²) = √9 = 3. If √(x²) were always x, we would get −3, which is negative and can't be the output of √. The correct identity is √(x²) = |x|: it equals x when x ≥ 0 and −x when x < 0.
Where does ±3 come from? The equation x² = 9 has two solutions, x = 3 and x = −3, written x = ±3. That is about solving an equation, not about the value of the √ symbol, which is always a single non-negative number.
This matters in JEE problems. For example, √(x² − 2x + 1) = √((x − 1)²) = |x − 1|, which splits into two cases at x = 1. In trigonometry and integration, √(1 − sin²θ) = |cos θ|, not cos θ; its sign depends on the quadrant. Note that (√x)² = x is fine, but only for x ≥ 0, where √x is defined as a real number.
Got this one? The Knot Scan finds the misconception you don't know you have, in 8 questions.