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Solving radical equations

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Solving Radical Equations

A radical equation has a variable inside a square root. Solve it by isolating the radical, squaring both sides, and solving the resulting equation -- then check every solution in the original equation, since squaring can introduce extraneous solutions that don't actually work.

Why radical equations need an extra step

A radical equation has a variable trapped inside a square root (or other radical). Solving one means squaring both sides to remove the radical — but squaring can introduce solutions that don't actually work in the original equation, called extraneous solutions. That's why the last step is always a check.

Solving a radical equation, with an extraneous-solution check Solve square root of (2x+3) = x. Step 1: radical is already isolated. Step 2: square both sides to get 2x+3 = x^2. Step 3: solve x^2-2x-3=0, factoring to (x-3)(x+1)=0, giving x=3 or x=-1. Step 4: check both in the original equation -- x=3 works since square root of 9 is 3, but x=-1 fails since square root of 1 is 1, not -1, so x=-1 is extraneous and rejected. 1 √(2x+3) = x (radical already isolated) 2 square both sides: 2x + 3 = x² 3 x² − 2x − 3 = 0 → (x−3)(x+1) = 0 → x = 3 or x = −1 4 check both in the ORIGINAL equation x = 3 √9 = 3 ∞ TRUE, keep it x = −1 √1 = 1 ≠ −1, extraneous
Squaring both sides can introduce a solution that fails the original equation.

The four-step method

  1. Isolate the radical on one side of the equation.
  2. Square both sides to eliminate the radical.
  3. Solve the resulting equation (often a linear or quadratic equation).
  4. Check every solution in the ORIGINAL equation — not the squared version.

Worked example

Solve √(2x + 3) = x. The radical is already isolated. Squaring both sides gives 2x + 3 = x², which rearranges to x² − 2x − 3 = 0, factoring to (x−3)(x+1) = 0, giving x = 3 or x = −1.

Checking in the original equation: for x = 3, √9 = 3, which is true — keep it. For x = −1, √1 = 1, but 1 ≠ −1 — this solution is extraneous and must be rejected. The only valid solution is x = 3.

Radical equations often build on earlier radical skills, like multiplying and dividing radicals to simplify a term before isolating it, or recognizing the shape of a square root function in the equation.

Why extraneous solutions appear

Squaring both sides of an equation is not a reversible step: it can turn a false statement (like 1 = −1) into a true one (1² = (−1)²). That's the mathematical root of the extraneous solution — and it's why radical functions and their equations always require a final check.

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