Extraneous Solutions
An extraneous solution is a value you get through valid algebra steps that still fails in the original equation, usually because squaring or clearing denominators created a wider equation than the one you started with.
Squaring both sides of an equation is not a reversible step. If a = b, then squaring gives a2 = b2, but the reverse is not guaranteed: a2 = b2 only tells you a = b or a = -b. So the squared equation accepts every solution of the original plus any new value that makes the two sides opposites instead of equal. Those extra values are extraneous: they solve the squared version but not the real one. The same kind of trap shows up with rational equations. Multiplying both sides by a denominator like (x - p) clears the fraction, but if the resulting candidate turns out to equal p, that denominator was zero all along, so the original expression was never defined there.
The fix is always the same: solve the transformed equation, then plug every candidate back into the original, untransformed equation before you trust it. A square root symbol only ever outputs a nonnegative number, so if your candidate makes the right side negative, reject it immediately. For rational equations, reject any candidate that makes a denominator zero.
This idea sits right behind quadratics and radical equations: solving a radical equation almost always means isolating the root, squaring, and landing on a quadratic with two roots, only one of which usually survives. It is the same vigilance you need with polynomial and rational equations, where factoring or cross-multiplying can quietly introduce a root the original expression cannot support.
Key terms
- Extraneous solution
- A value produced by correct algebra on a transformed equation that does not satisfy the original equation.
- Squaring both sides
- An operation that turns a = b into a2 = b2, which also accepts cases where a = -b.
- Domain restriction
- A value excluded from an equation's valid inputs, such as one that makes a denominator zero or a square root negative.
- Clearing denominators
- Multiplying every term by the denominators present to turn a rational equation into a polynomial one.
Watch it explained
- How it worksHow does squaring both sides widen the pool of solutions instead of preserving it exactly?
- Worked exampleSolve the equation where the square root of (3x+10) equals x, and show which candidate gets rejected.
- Common mistakeWhy isn't checking your answers optional the moment a radical or a denominator shows up?
- Going furtherWhat operations besides squaring can conjure solutions that were never in the original equation?
Check yourself
End of topic.