करणी समीकरण समाधानकर्ता

Extraneous solutions are values produced algebraically that do not satisfy the original equation. They occur because raising both sides to an even power (like squaring) is not reversible — it can create new solutions that the original equation never had. Always plug answers back into the original to verify.

Squaring both sides transforms "f(x) = g(x)" into "f(x)² = g(x)²", which is equivalent to "(f(x)−g(x))(f(x)+g(x)) = 0". This added factor introduces solutions from f(x) = −g(x) that were never part of the original. Checking filters these out.

Cube roots accept all real radicands (positive, negative, or zero), so there is no domain restriction. Cubing both sides is a bijective (one-to-one) operation over the reals, meaning it introduces no extraneous solutions. Square roots require the radicand ≥ 0, and squaring is not injective, so extraneous solutions are possible.

For √(ax + b), the radicand must satisfy ax + b ≥ 0, giving x ≥ −b/a (when a > 0) or x ≤ −b/a (when a < 0). Any solution outside this domain is extraneous regardless of whether the algebra is otherwise correct.

An equation like √(ax+b) = cx+d becomes, after squaring, the quadratic c²x² + (2cd−a)x + (d²−b) = 0. Apply the quadratic formula: x = [−(2cd−a) ± √((2cd−a)² − 4c²(d²−b))] / (2c²). Both roots must be checked in the original. Also verify the right-hand side cx+d ≥ 0 (since √ outputs a non-negative value).

Nested radical equations have a radical inside another radical, e.g. √(a + √(bx+c)) = d. The method is to peel layers outward: square the outer equation first to reveal and isolate the inner radical, then square again to eliminate it. Each squaring step may introduce extraneous solutions, making the verification step especially important.

Yes. If all candidate solutions turn out to be extraneous, the equation has no real solution. For example, √(x+5) = −3 has no solution because square roots are always non-negative and can never equal −3. The algebra would produce x = 4, but checking: √(4+5) = √9 = 3 ≠ −3, so it is extraneous and the solution set is empty.