How the quadratic formula works
Any equation of the form a·x² + b·x + c = 0 (with a ≠ 0) can be solved directly with the quadratic formula, x = (−b ± √(b²−4ac)) / (2a). The two signs of the square root give the two roots, which are the x-values where the parabola crosses the horizontal axis.
You do not need to factor or complete the square — plug the three coefficients in and read off the answers. This calculator keeps full precision internally and rounds only for display.
What the discriminant tells you
The quantity under the square root, b²−4ac, is called the discriminant, and its sign determines what kind of roots you get before you compute anything else.
- Positive discriminant: two distinct real roots (the parabola crosses the axis twice).
- Zero discriminant: one repeated real root (the parabola just touches the axis).
- Negative discriminant: a complex-conjugate pair, written a ± b·i (the parabola never touches the axis).
Common mistakes to avoid
Watch the signs: b and c are often negative, and a stray sign is the most frequent error. Make sure the equation is in standard form with everything moved to one side and zero on the other before reading off a, b and c.
If a is zero the equation is linear, not quadratic, so the formula does not apply — this tool flags that case rather than dividing by zero.
Formula
disc = b²−4ac; x = (−b ± √disc) / (2a)Frequently asked questions
- What does the discriminant tell me?
- The value b²−4ac decides the roots: positive gives two distinct real roots, zero gives one repeated root, and negative gives a complex-conjugate pair.
- Can this solve equations with no real solutions?
- Yes. When the discriminant is negative the calculator returns the complex roots rather than reporting a failure.
- What if a is zero?
- Then the equation is linear, not quadratic, and the formula divides by zero. Enter a non-zero value for a; solve bx + c = 0 by hand instead.