A radical equation contains a variable inside a square root, and it needs one extra step that other equations do not: a final check of every answer. The solving itself is short. You can check your work with the free tools on Calc Solver.
What a Radical Equation Is
A radical equation has the variable under a radical sign, such as sqrt(x + 5) = 3 or sqrt(2x – 1) + 2 = 7. The goal is the same as always: find the value of x that makes the equation true. The radical only changes the first moves.
Step 1: Isolate the Radical
Get the square root alone on one side before doing anything else. For sqrt(2x – 1) + 2 = 7, subtract 2 from both sides to get sqrt(2x – 1) = 5. If two radicals appear, isolate one of them first and repeat the process later.
Step 2: Square Both Sides
Square both sides to remove the root: (sqrt(2x – 1))^2 = 5^2 gives 2x – 1 = 25. Squaring is the step that creates the risk of extra answers, so remember that the check at the end is not optional.
Step 3: Solve the New Equation
Solve the ordinary equation that remains: 2x – 1 = 25 gives 2x = 26, so x = 13. If squaring produced a quadratic, solve it by factoring or the quadratic formula, which may give two candidates instead of one.
Step 4: Check Every Answer
Put each candidate back into the ORIGINAL equation, never only into the squared version. For x = 13: sqrt(2 times 13 – 1) + 2 = sqrt(25) + 2 = 5 + 2 = 7, correct. Any candidate that fails this check is extraneous and must be thrown out, because squaring can turn a false statement into a true one.
Common Mistakes to Avoid
First, squaring before isolating the radical, which leaves a mess. Second, skipping the check and keeping an extraneous answer. Third, forgetting that a square root result is never negative, so an equation like sqrt(x) = -4 has no solution at all.
Practice Problems to Try
Solve sqrt(x + 3) = 4, then sqrt(3x – 2) – 1 = 4, then sqrt(x) + sqrt(x + 5) = 5 (isolate one radical, square, repeat). Check every answer in the original equation.