Eleventh Grade · Math · week 25

Solving radical equations

Solve radical equations and check for extraneous solutions introduced by squaring.

To solve an equation with a variable under a square root, isolate the radical on one side, then square both sides to eliminate it -- squaring is how you "undo" a square root. This turns the equation into a polynomial equation you can solve with familiar tools (usually factoring).

Squaring both sides is NOT a fully reversible move: it can turn a false statement into a true one (since both a positive and a negative number square to the same positive result), which is exactly how extraneous solutions sneak in. Every solution you find must be substituted back into the ORIGINAL radical equation -- not the squared version -- to confirm it actually works.

📋 For the teacher
Teach it (10–15 min):
  • Insist on isolating the radical completely before squaring -- squaring a side with extra terms attached to the root creates a much messier (and error-prone) expansion.
  • Make the final check a fixed, unskippable ritual: substitute back into the ORIGINAL equation, every single time, no exceptions.
  • Use the sqrt(x)=-3-style example early to build the instinct that a principal square root is never negative -- some equations have no solution before you even square anything.
  • Compare this week's "check for extraneous solutions" habit explicitly to the rational-equation domain checks from a few weeks ago -- it is the same underlying caution, applied to a new kind of equation.
Talk about it:
  • Why does squaring both sides of an equation risk introducing a false solution?
  • What must be true about the non-radical side of a radical equation for a solution to be valid?
  • Why does sqrt(x) = -3 have no solution, without any algebra at all?

Mastery looks like: They isolate and square correctly to solve a radical equation, and reject any extraneous solution by checking it in the original equation.

Common stumbles: Squaring before fully isolating the radical; skipping the final check and reporting an extraneous solution as valid.

Homework for this lessonType it, write it with a stylus, or print it for pencil & paper.open →
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