Quadratic Formula Calculator
Solve ax² + bx + c = 0 using the quadratic formula and learn whether the roots are real or complex.
Quadratic equation solver
The formula is x = (-b ± √(b² - 4ac)) / (2a). If the discriminant is negative, the solutions are complex numbers.
Discriminant:
Root 1:
Root 2:
Equation type:
Formula used:
What the discriminant tells you
The discriminant (b² − 4ac) determines the nature of the roots before you even compute them: a positive discriminant means two distinct real roots, a discriminant of exactly zero means one repeated real root (the parabola just touches the x-axis), and a negative discriminant means two complex (non-real) roots, since it would require taking the square root of a negative number.
Frequently asked questions
What do complex roots mean graphically? They mean the parabola never crosses the x-axis — it stays entirely above or entirely below it, so there's no real value of x where the equation equals zero.
What happens if "a" is zero? The equation is no longer quadratic — it becomes linear (bx + c = 0), which the calculator detects and solves directly as x = −c/b instead of applying the quadratic formula.
Why does the formula have a "±" symbol? It compactly represents both possible roots at once — adding the square root term gives one root, and subtracting it gives the other, which is why a quadratic equation generally has exactly two solutions.