Shortcuts when both sides are always non-negative

The splitting into cases arose from the fact that the right-hand side could have any sign. If instead the right-hand side is, by construction, a quantity that is always $\ge 0$ (another radical, an absolute value, a distance), then the sign study is free: it is enough to impose the existence conditions and square directly, without distinguishing cases.

• Squaring without splitting into cases
• Example — distance in a pencil of lines
• Example — two radicals compared