Isolate and Square

Algebra 1 · Axiom Academy

A reliable strategy for solving radical equations — get the root alone, square it away, then check what survives. A square root and a square are inverse operations: squaring a root cancels it exactly . So the plan is to get the radical by itself, then apply a square to both sides — the root peels away and what's left is a polynomial you already know how to solve. Solve . The radical already sits alone on the left, so we can square straight away. Watch the root lift off, the polynomial solve out, and the answer ride back into the original to be checked. 3. When the Radical Isn't Alone Yet Solve . Here a stray +2 shares the side with the root, so the radical is not isolated. Strip the +2 off first; only then is squaring clean. The animation peels the +2 away before the square ever lands. 4. The Danger: Extraneous Solutions This is why Step 4 is non-negotiable. Solve . Squaring produces a quadratic with two candidate roots — but only one of them actually satisfies the original. The animation drops each candidate back in and lets one pass and one fail. Squaring is not reversible: two different numbers can square to the same value, so squaring can fold a new solution in that the original never had. Watch how -3 gets folded onto the very same square as 3 . Both sides agree when you substitute — keep it. The sides disagree (often a root forced to equal a negative) — reject it as extraneous.

This is the written version of the interactive lesson above. See the full Algebra 1 course.