Equation Solver

This solver supports three types: linear equations (ax + b = c), quadratic equations (ax² + bx + c = 0), and cubic equations (ax³ + bx² + cx + d = 0). Enter the coefficients for your chosen type and the solver finds all real roots.

The discriminant (Δ = b² − 4ac for quadratics) tells you how many real roots an equation has. If Δ > 0, there are two distinct real roots; if Δ = 0, there is exactly one real root (a repeated root); if Δ < 0, there are no real roots (the roots are complex/imaginary).

For quadratic equations where the discriminant is negative, the roots are complex (imaginary). This solver displays real roots only. When no real root exists, the result will indicate that the roots are complex.

For a linear equation ax + b = c, rearrange it to ax + (b − c) = 0 form. So 2x + 5 = 11 becomes 2x − 6 = 0. Enter a = 2, b = −6, c = 0 under the Linear type. The solver gives x = 3.

Yes. All coefficient fields accept decimal values. For example, you can enter a = 0.5, b = 1.25, c = −3.75 and the solver will compute accurate decimal roots.

The cubic solver uses the depressed cubic method and numerical techniques to find up to three real roots for equations of the form ax³ + bx² + cx + d = 0. All three roots (where real) are displayed along with a verification table.

If a = 0 in a quadratic or cubic, the equation reduces to a lower degree. For example, a quadratic with a = 0 becomes linear. The solver handles this gracefully — if a = 0 is entered for a quadratic, it falls back to solving bx + c = 0.

The solution verification table shows each root substituted back into the equation, confirming f(x) ≈ 0. Small rounding differences (e.g. 0.000001) are normal due to floating-point precision and confirm the root is correct.