What is a radical equation?
A radical equation is an equation with the variable inside a root, such as √(2x + 1) = x − 1. An equation like x = √5 doesn't count, since the root contains only a number.
To enter one, type sqrt( ) around the expression under the root, or paste the √ symbol with brackets. Both sqrt(x + 3) = x - 3 and √(x + 3) = x - 3 work.
What are the steps for solving radical equations?
Isolate, square, solve, check. The last step is required, not optional, because squaring can add answers that don't work.
- Move everything except the square root to the other side.
- Square both sides. The root disappears on the left.
- Multiply out the right side and collect everything on one side.
- Solve the result, usually by factoring or the quadratic formula.
- Substitute each answer into the original equation and keep only the ones that work.
The final answer is x = 6. The value x = 1 is extraneous: it solves the squared equation but not the original one.
Why does squaring create extraneous solutions?
Squaring hides the sign of each side. Two numbers that are different can have the same square, so the squared equation accepts answers the original rejects.
The simplest case is −2 = 2. That's false, but squaring both sides gives 4 = 4, which is true. The same thing happened above with x = 1: the left side was 2, the right side was −2, and their squares matched.
A principal square root, written √, is never negative. So any candidate that makes the other side negative is extraneous. That's a quick way to spot them before you finish the arithmetic.
Why do you isolate the square root before squaring?
If another term sits next to the root, squaring doesn't remove the root. Instead you get a messier equation with the root still inside a middle term.
For √(x + 7) + 5 = x, subtract 5 first to get √(x + 7) = x − 5. Squaring gives x + 7 = x² − 10x + 25, so x² − 11x + 18 = 0 and x = 2 or x = 9. The check rejects x = 2, since √9 = 3 while 2 − 5 = −3. The answer is x = 9.
A number multiplied by the root is fine to leave in place. For 2√(x + 1) = x − 2, squaring both sides gives 4(x + 1) = x² − 4x + 4, so x² − 8x = 0 and x = 0 or x = 8. At x = 0 the left side is 2 and the right side is −2, so the solver keeps only x = 8. The solver shows exactly these steps.
How do you square the other side correctly?
Treat it as a binomial times itself: (a − b)² = a² − 2ab + b². The middle term is the one people forget.
| Side to square | Squared correctly | Common mistake |
|---|---|---|
| x − 3 | x² − 6x + 9 | x² + 9 or x² − 9 |
| x − 1 | x² − 2x + 1 | x² + 1 |
| x − 5 | x² − 10x + 25 | x² + 25 |
| 2√(x + 1) | 4(x + 1) | 2(x + 1) |
The last row is the other trap. When a number multiplies the root, square the number too. For √(2x + 1) = x − 1, squaring correctly gives 2x + 1 = x² − 2x + 1, so x² − 4x = 0. That gives x = 0 or x = 4, and the check leaves x = 4.
When does a radical equation have no solution?
It has no solution when every candidate fails the check, or when the isolated root equals a negative number. A principal square root can't be negative, so √(something) = −2 is impossible.
Take √x + 2 = 0. Subtracting 2 gives √x = −2, and the calculator stops right there with No solution. Squaring anyway would give x = 4, which fails the check since √4 = 2 and 2 + 2 is 4, not 0.
What kinds of radical equations can this solver handle?
The radical equation solver handles one square root in an equation, with a linear or quadratic expression on the other side. After squaring, it solves the result the same way the quadratic equation solver does, so answers like x = (−1 + √17)/2 for √(x + 5) − 1 = x come out exact.
You can use any letter for the variable, so √(t + 3) = t − 3 works the same way. Every answer comes with the squared equation, the candidates it produced, and a check line for each candidate that says whether it was kept or rejected and why.
Two things are outside its range. Equations with two or more square roots, such as √x + √(x + 5) = 5, aren't supported yet. Cube roots inside an equation, like ∛(x + 1) = 2, aren't either. For simplifying a single radical such as √72, use the simplify radicals calculator instead.