Radical Equations

Squaring undoes a square root: √n² = n. That only works cleanly if the radical is alone on one side first. If other terms are mixed in with it, squaring the whole side creates a mess of cross terms instead of canceling the root. So the first move, always, is to get the radical by itself — move everything else to the other side before you square anything.

Once the radical stands alone, square both sides. The square root vanishes and you are left with a linear or quadratic equation you already know how to solve. Solve that equation normally, then go back to the original radical equation and plug each answer in. Squaring can introduce extraneous solutions — numbers that satisfy the squared version but make the original square root equal a negative number, which is impossible since a square root symbol always means the nonnegative root. Any such answer gets thrown out.

When two radicals both carry the variable, you cannot isolate both at once. Isolate one, square once, and you still have a radical left — so isolate that one and square again. Each squaring step is a chance to introduce extraneous roots, so the final check matters even more. This topic sits next to quadratics because squaring a linear expression under a root often produces a quadratic to factor or solve with the quadratic formula, linking radical equations directly to that skill.