Why Is the Square Root of x Squared Equal to |x|?

Is √(x²) always x?

For a real number x, √(x²) = |x|, the nonnegative magnitude of x. It equals x when x is nonnegative and −x when x is negative. The square-root symbol means the principal, nonnegative square root, so it cannot produce a negative result here.

Check a negative input

Let x = −5. Squaring gives x² = 25, and √25 = 5. Simply cancelling the square and square root would incorrectly give −5. The absolute-value form works: |−5| = 5.

If x = 4, the same rule gives √16 = |4| = 4. At x = 0, both sides are zero. Thus a single absolute-value expression handles positive, negative and zero inputs correctly.

Why does x² = 25 have two solutions?

The equation asks which real numbers become 25 when squared. Both 5 and −5 do, so x = ±5. In contrast, the expression √25 denotes just 5. Distinguish evaluating a square root from solving an equation containing a square.

Similarly, √((x − 3)²) = |x − 3|. When x is at least 3, this becomes x − 3. When x is less than 3, it becomes 3 − x. For x = 1, the value is 2, not −2.

A useful error check

For √(9x²), write √((3x)²) = |3x| = 3|x|. Writing 3x is valid only after establishing x ≥ 0. Do not assume an unknown variable is positive just because a diagram or earlier example used positive numbers.

Practice with answers

• √((-8)²) = 8.
• √((2 − 7)²) = 5.
• Solve y² = 49: y = 7 or y = −7.
• Simplify √((t + 2)²) for t < −2: −t − 2.

Substitute a permitted negative value into a proposed simplification. If one side becomes negative while the square root remains nonnegative, the simplification needs correction.

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