Quadratic Formula Solver

It means the discriminant (b squared minus 4ac) came out negative, so the parabola described by your equation never crosses the x-axis — there is no real number x that makes the equation equal zero. The equation does have solutions, but they involve the imaginary unit i, which this calculator doesn't compute since Root 1 and Root 2 are meant to represent real x-axis intersections.

When a is zero, the x² term vanishes and the equation ax² + bx + c = 0 simplifies to the linear equation bx + c = 0, which this solver detects and solves directly as x = -c/b instead of attempting to divide by zero in the quadratic formula's denominator. Both Root 1 and Root 2 will show the same single value in this case, since a linear equation has only one solution.

They come from the plus and minus branches of the ± in the quadratic formula, representing the two points where the parabola crosses the x-axis, and they're mathematically interchangeable — neither is more 'correct' than the other. The two roots are always symmetric around the parabola's vertex at x = -b/2a, equally spaced on either side of it.

That happens when the discriminant equals exactly zero, meaning the parabola's vertex sits precisely on the x-axis, touching it at a single point rather than crossing through two distinct points. This is called a repeated or double root, and the Roots field will read 'one-real' to flag that special case.