Radical Equation Calculator
This radical equation calculator solves for x under a square or cube root, keeps the answer exact, and tells you why a rejected candidate was rejected rather than quietly dropping it.
Type sqrt(x + 7) = x - 5 and you get x = 9 — with x = 2 shown beside it and the reason it cannot be right.
Four Steps, and the Fourth Is Not Optional
Isolate the root. Raise both sides to the power. Solve what is left. Test every answer in the equation you started with.
Most people can do the first three. The fourth is where the marks live, because squaring is not a reversible move — it turns −3 into 9 exactly as it turns +3 into 9. An answer that made one side negative can sail through the squaring without ever having solved the original.
Watch It Happen
Take √(x + 7) = x − 5. Square both sides and you get x + 7 = x² − 10x + 25, so x² − 11x + 18 = 0, so (x − 2)(x − 9) = 0.
Two candidates: x = 2 and x = 9.
Check x = 9. Left: √16 = 4. Right: 9 − 5 = 4. They match, so 9 is a real answer.
Check x = 2. Left: √9 = 3. Right: 2 − 5 = −3. And there it is — a square root is never negative, so 3 can never equal −3. The number 2 solves the squared equation and has nothing to do with the one you were given.
That reason is worth writing down in an exam. "x = 2 is extraneous" gets you less than "x = 2 gives √9 = 3 but x − 5 = −3, and a square root cannot be negative".
The Answer Is Often a Surd
√(2x + 5) = x squares to x² − 2x − 5 = 0, and the quadratic formula gives x = 1 ± √6.
One of those survives: 1 + √6. The other, 1 − √6, is about −1.449, and a negative x cannot equal a square root.
Most calculators hand you 3.449 and stop there. That figure is what 1 + √6 looks like rounded, not the thing itself, and a marking scheme will usually say so. Nothing on this page is rounded before you see it.
| Equation | Candidates | Survives |
|---|---|---|
| √(3x + 1) = 4 | 5 | x = 5 |
| √(x + 7) = x − 5 | 2, 9 | x = 9 |
| √(2x + 5) = x | 1 ± √6 | 1 + √6 |
| √x = −5 | 25 | none |
Sometimes Nothing Survives
√x = −5 is the shortest example in the subject: square it, get x = 25, check it, and find √25 is 5 rather than −5.
So nothing works. And you could have seen it before squaring: a square root on the left, a negative on the right, two things that never meet. Write it down and stop.
Cube Roots Behave Differently
Everything above is about even roots. Odd roots follow different rules and the difference is worth holding on to.
∛(x − 1) = −2 is perfectly ordinary. Cube both sides: x − 1 = −8, so x = −7. Check it: ∛(−8) = −2. Correct, and no candidate needed throwing away.
That is not luck. An odd root is defined for every real number, negative ones included, and it keeps the sign of whatever sits inside it. Raising to an odd power is a reversible move, so it has nothing to invent with. An odd-index radical equation has no extraneous solutions at all — the check still costs you ten seconds, but it cannot fail for that reason.
Two Conditions, Worth Writing Before You Start
For an even root, the equation itself tells you two things before any algebra happens.
What sits under the root cannot be negative. And the other side cannot be negative either, because that is what the root produces.
For √(x + 7) = x − 5 those read: x + 7 ≥ 0, so x ≥ −7; and x − 5 ≥ 0, so x ≥ 5. Between them they rule out everything below 5 — which disqualifies x = 2 before you have squared anything at all. This page prints both conditions above the working, because half the time they tell you the answer early.
How to Use This Radical Equation Calculator
Tap √ for a square root or the cube root key beside it, and put what is underneath inside the brackets: sqrt(2x + 5) = x. A number in front works and so does one added on, as in 2sqrt(x - 1) + 3 = 9 — those get tidied away first.
Every candidate is tested in the equation you typed, and any that fails is shown with its reason. If your equation has two roots in it, that needs squaring twice and is beyond this page. If the root is being simplified rather than solved, that is simplify radicals, and if x sits on the line rather than under a root, you want the linear equation page. Once the root is gone you are usually left with a quadratic, and the quadratic formula finishes it.
Radical Equation Calculator FAQ
What are the steps for a radical equation?
What is an extraneous solution?
Why does √(x + 7) = x − 5 reject x = 2?
Is it possible for nothing at all to work?
Do cube root equations have extraneous solutions?
Surd or decimal — which do I write?
What conditions must hold before I solve?
How do I handle an equation with two roots in it?
Isolate, raise, solve — then put every answer back and make it prove itself.
