Radical Equation
A radical equation is an equation in which the variable appears inside a radical (such as a square root, cube root, or other root). To solve one, you typically isolate the radical and then raise both sides to a power that eliminates it.
A radical equation is an algebraic equation in which at least one term contains a variable expression under a radical symbol. Solving such equations involves isolating the radical on one side, then raising both sides of the equation to the index of the radical to remove it. Because this process can introduce values that satisfy the transformed equation but not the original, all solutions must be checked in the original equation to identify and discard any extraneous solutions.
Worked Example
Problem: Solve the equation .
Step 1 – Isolate the radical: The radical is already isolated on the left side, so no rearranging is needed.
Step 2 – Square both sides: Raise both sides to the power of 2 to eliminate the square root.
Step 3 – Solve the resulting equation: Subtract 3 from both sides, then divide by 2.
Step 4 – Check for extraneous solutions: Substitute back into the original equation to verify it works.
Answer:
Why It Matters
Radical equations appear frequently in physics and engineering — for instance, the formula for the period of a pendulum involves a square root, and solving for the pendulum's length requires handling a radical equation. They also give you essential practice with a broader algebraic skill: reversing operations (in this case, undoing a root by raising to a power) while watching for solutions that don't actually work.
Common Mistakes
Mistake: Forgetting to check for extraneous solutions
Correction: Squaring (or raising to any even power) can introduce false solutions. Always substitute your answers back into the original equation. If a value makes the original equation false, discard it.
Mistake: Squaring both sides before isolating the radical
Correction: If the radical is not isolated first, squaring both sides creates a more complicated equation and often leads to errors. Move all non-radical terms to the other side before raising to a power.