Radical Equation Solver

Solve equations containing radicals step by step with the Radical Equation Solver. Enter your radical equation (e.g. sqrt(x+3) = 5 or sqrt(2x-1) = x-2) and choose the index of the radical. You get back the solution(s), a check for extraneous roots, and a verification showing whether each answer satisfies the original equation.

Select the form that matches your equation.

Index of the radical — 2 for square root, 3 for cube root, etc. Only used for the 'custom index' form.

Coefficient of x inside the radical.

Constant term inside the radical.

For forms √(ax+b)=c, this is c. For √(ax+b)=x+c, this is c. For √(ax+b)=√(cx+d), this is c.

Used in forms that have a second radical or an additional constant (e.g. √(cx+d) or +cx=d).

Results

Solution(s) for x

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Extraneous Solution Detected

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LHS Verification (at solution)

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RHS Verification (at solution)

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Number of Valid Solutions

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Results Table

Frequently Asked Questions

What is a radical equation?

A radical equation is any equation in which the variable appears inside a radical symbol — most commonly a square root (√), but also cube roots (∛) or nth roots. For example, √(x + 3) = 5 and ∛(2x − 1) = 4 are both radical equations. The key challenge is isolating the radical and then raising both sides to the appropriate power to eliminate it.

How do you solve a radical equation step by step?

First, isolate the radical expression on one side of the equation. Then raise both sides to the power equal to the radical's index (square both sides for a square root, cube both sides for a cube root, etc.). This produces a polynomial equation you can solve normally. Finally — and critically — substitute every candidate solution back into the original equation to discard any extraneous roots.

What are extraneous solutions and why do they appear?

Extraneous solutions are values of x that satisfy the transformed (squared or raised-power) equation but do NOT satisfy the original radical equation. They arise because raising both sides to an even power can introduce solutions that were never valid (for example, squaring both sides of √x = −3 gives x = 9, but substituting back shows √9 = 3 ≠ −3). Always check every candidate in the original equation.

Why do we care about radical equations in real life?

Radical equations appear in physics (e.g. calculating the period of a pendulum: T = 2π√(L/g)), engineering (stress and strain formulas), geometry (the Pythagorean theorem and distance formulas), and finance (geometric mean return calculations). Knowing how to solve them accurately is essential in any field involving rates, distances, or power laws.

Can a radical equation have no solution?

Yes. If all candidate solutions turn out to be extraneous after the verification step, the equation has no real solution. This typically happens when the original equation sets a radical (which is always non-negative for even indices) equal to a negative number, or when the domain of the radicand excludes the candidate values.

How do you handle equations with two radicals?

When an equation has two radical expressions, isolate one radical on one side, then raise both sides to the appropriate power. This may leave a second radical in the resulting equation; repeat the process — isolate the remaining radical and raise both sides again. After clearing all radicals, solve the resulting polynomial and check all candidates for extraneous roots.

What common mistakes should I avoid when solving radical equations?

The most common mistakes are: (1) forgetting to check for extraneous solutions after squaring, (2) squaring only the radical rather than the entire side of the equation (e.g. writing (√(x+3))² = c instead of squaring the full RHS when the RHS has multiple terms), and (3) ignoring domain restrictions — the expression inside an even-index radical must be non-negative.

Does the radical index affect how many solutions there can be?

Yes. For even indices (square root, 4th root, etc.), raising both sides to the power can introduce extraneous solutions and the radicand must be ≥ 0. For odd indices (cube root, 5th root, etc.), the radical is defined for all real numbers and raising to the odd power does NOT introduce extraneous solutions, so verification is simpler though still good practice.

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