Solving Radical Equations

Calculus Readiness · Axiom Academy

LESSON Solving Radical Equations Isolate the radical, raise both sides to undo it, solve — then check, because squaring can invent solutions that were never there. Before you can undo a root, the root has to stand alone on one side. Anything added to it gets moved across the equals sign first. Take : subtract the 1 from both sides and the radical is isolated, ready to be squared. Move the + 1 across the equals sign… 2. Square Both Sides to Undo the Root Squaring is the inverse of a square root: peels the radical off and leaves the radicand behind. Watch the root lift away from — the left collapses to 3x+4 while the right becomes 25 . Then it is just a linear equation. Whatever you do to one side you do to the other — square the left and the right together. . Squaring exactly undoes the square root. 5^2 = 25 . A plain number on the far side is simply raised to the same power. Square for , cube for — raise to whatever the index is. From 3x+4=25 : subtract 4 to get 3x=21 , then divide by 3 . So x=7 . Checking in the original: . It works. 3. Check — and Catch the Fake Root Squaring can invent solutions, so every candidate must be tested in the original equation. Solve : squaring gives x=(x-2)^2 , i.e. x^2-5x+4=0 , which factors as (x-1)(x-4)=0 — candidates x=1 and x=4 . Now substitute each one back and watch what happens. x=4 passes: and 4-2=2 . Both sides equal 2 — keep it. x=1 fails: but 1-2=-1 . Since , reject it. It is an extraneous root.

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