Quadratic in form solver

Enter the three coefficients and choose whether u represents x squared or the principal square root of x. The solver finds the substituted roots, applies the required domain restriction, and then back-substitutes to return every real value of x. It also displays the discriminant and a compact sequence of steps, making the result useful for checking homework, preparing examples, or validating an algebra workflow.

Recognize a quadratic hidden inside another expression

An equation is quadratic in form when the same expression appears once to the first power and once squared. A biquadratic equation such as x to the fourth minus five x squared plus four equals zero contains x squared and its square, x to the fourth. Setting u equal to x squared changes it into u squared minus five u plus four equals zero. The same pattern occurs with radicals: if an equation contains x and the square root of x, then x is the square of that root, so setting u equal to the square root of x produces an ordinary quadratic. This solver asks for coefficients a, b, and c in the transformed pattern a times u squared plus b times u plus c equals zero. Choose the substitution that matches the original expression. This separation is valuable because it keeps the familiar quadratic calculation distinct from the later domain and back-substitution work, where solutions are most often lost or added incorrectly.

Solve for u, then enforce its domain

After substitution, the solver computes the discriminant b squared minus four ac. A negative discriminant means there are no real values of u and therefore no real values of x for the supported forms. A zero discriminant gives one repeated u value, while a positive discriminant gives two candidates. Those candidates are not automatically valid. Both supported substitutions produce a nonnegative u: a real square x squared cannot be negative, and the principal square root of x is also never negative. The solver therefore reports all quadratic roots in u_solutions and separately reports the roots that survive in admissible_u_solutions. This makes a rejected candidate visible instead of silently discarding it. The calculation uses a numerically stable form of the quadratic formula when the roots differ, reducing avoidable cancellation for coefficients of very different sizes. Coefficients must be finite real numbers, and a must be nonzero because otherwise the transformed equation is linear rather than quadratic in form.

Back-substitute without missing or inventing roots

Back-substitution depends on what u means. When u equals x squared, each positive admissible u creates two real answers: positive and negative square root of u. A zero u creates only x equals zero, not two distinct copies. This is why a fourth-degree equation can have four, two, one, or no distinct real solutions even though its substituted quadratic has at most two roots. When u equals the principal square root of x, solving square root of x equals u gives x equals u squared. Only one x comes from each admissible u, because the radical denotes the nonnegative principal root. The solver sorts and deduplicates the final values and returns solution_count alongside them. You should still compare the chosen substitution with the structure of the original equation before using the result. The tool solves the declared quadratic form; it does not parse arbitrary algebraic notation or infer coefficients from a typed equation. For automation, the same deterministic result is available for $0.002 per request.