Quadratic Formula Calculator
Solve quadratic equations of the form ax² + bx + c = 0.
What the discriminant tells you before you solve
A quadratic is ax² + bx + c = 0 with a ≠ 0, and the formulax = (−b ± √(b² − 4ac)) / 2a solves all of them. Before computing anything, though, thediscriminant b² − 4ac tells you what kind of answer to expect: positive gives two distinct real roots, zero gives one repeated root, and negative gives two complex conjugates and no real solution.
Geometrically that is where the parabola meets the x-axis — twice, once at a tangent, or not at all. The sign of a decides the direction it opens, upward when positive and downward when negative, and the vertex sits at x = −b / 2a, which is also the axis of symmetry and the location of the minimum or maximum.
Factoring is faster when it works, and completing the square is what the formula is derived from — worth seeing once, because it explains where every term comes from. Two checks on any answer: the roots should sum to −b/a and multiply to c/a. In applied problems, discard roots that are mathematically valid but physically impossible, such as a negative time or length.
Frequently Asked Questions
What is the quadratic formula?
x = (-b ± √(b² - 4ac)) / (2a), used to solve equations of the form ax² + bx + c = 0.
What does the discriminant (b² - 4ac) indicate?
If b² - 4ac > 0, there are 2 distinct real roots. If = 0, there is 1 repeated real root. If < 0, there are 2 complex conjugate roots.
What is the quadratic formula?
x = (-b +/- sqrt(b^2 - 4ac)) / 2a, solving any equation of the form ax^2 + bx + c = 0 where a is not zero. It comes from completing the square on the general form, so it always works even when factoring is impractical.
What does the discriminant tell me?
The discriminant b^2 - 4ac determines the nature of the roots: positive gives two distinct real roots, zero gives one repeated real root where the parabola touches the axis, and negative gives two complex conjugate roots and no x-intercept. Checking it first tells you what kind of answer to expect.