📐 Quadratic Equation Calculator
Solve any quadratic equation of the form ax² + bx + c = 0. Enter the coefficients a, b and c to get both roots, including real and complex solutions, with a full discriminant breakdown.
What is this tool?
A quadratic equation is any equation that can be written in the form ax² + bx + c = 0, where a, b and c are real numbers and a is not zero. Quadratic equations appear throughout mathematics, physics, engineering and economics. The path of a thrown ball, the shape of a satellite dish, the break-even point in a business model and the trajectory of a projectile all involve quadratics.
The solutions to a quadratic equation are called roots, and every quadratic has exactly two roots (though they may be identical or complex). The most powerful method for finding them is the quadratic formula: x = (−b ± √(b² − 4ac)) / 2a. The expression under the square root, b² − 4ac, is called the discriminant and it tells you what kind of roots to expect before you even calculate them.
This quadratic equation calculator lets you enter the coefficients a, b and c and instantly get both roots. If the discriminant is positive, you get two distinct real roots. If it is zero, you get one repeated real root. If it is negative, the roots are complex numbers. The tool shows each step of the calculation so you can follow along and verify your work.How it works
The quadratic formula is derived by completing the square on the general form ax² + bx + c = 0. The result is x = (−b ± √(b² − 4ac)) / 2a. First the calculator computes the discriminant Δ = b² − 4ac. If Δ is positive, it takes the square root and evaluates both the plus and minus branches to get two real roots. If Δ is zero, both branches give the same value x = −b / 2a, which is the vertex of the parabola sitting on the x-axis. If Δ is negative, the square root of a negative number produces an imaginary component, and the roots are x = −b / 2a ± Δ_abs·i / 2a, where i is the imaginary unit.
The calculator also determines the vertex of the parabola y = ax² + bx + c, located at x = −b / 2a, and whether the parabola opens upward (a > 0) or downward (a < 0). This gives you a complete picture of the equation and its graph.How to use
- Enter the coefficient a (must not be zero).
- Enter the coefficient b.
- Enter the coefficient c.
- Click Calculate to see both roots and the discriminant.
- Check whether the roots are real, repeated or complex.
Frequently Asked Questions
What happens if a is zero?
If a equals zero, the equation is no longer quadratic but linear (bx + c = 0), which has a single root x = −c / b. The calculator requires a to be non-zero; if you enter zero, it will prompt you to use a linear equation solver instead.
What are complex roots?
When the discriminant b² − 4ac is negative, the equation has no real-number solutions. Instead, the roots are complex numbers of the form p + qi, where i is the imaginary unit (√−1). Complex roots always come in conjugate pairs and indicate that the parabola does not cross the x-axis.
Can this calculator factor the quadratic?
Factoring is a separate method that works when the roots are rational numbers. This calculator uses the quadratic formula, which works for all quadratics regardless of whether the roots are rational, irrational or complex. If you need factoring specifically, use the Factoring Calculator.
What is the discriminant?
The discriminant Δ = b² − 4ac tells you the nature of the roots without computing them. Positive Δ means two distinct real roots. Zero Δ means one repeated real root (the parabola touches the x-axis at exactly one point). Negative Δ means two complex conjugate roots (the parabola does not intersect the x-axis).
Tips & Advice
When solving quadratic equations by hand, always start by computing the discriminant b² − 4ac before plugging into the quadratic formula. If the discriminant is a perfect square (like 25 or 144), the roots will be rational and you could have factored instead. If the discriminant is zero, you only need to compute one value (−b / 2a), which is both roots. In physics problems, pay attention to which root is physically meaningful: for projectile motion, the positive root usually gives the landing time, while the negative root is discarded. In optimization problems, x = −b / 2a gives the location of the vertex, which is either the minimum (a > 0) or the maximum (a < 0) of the quadratic function. Always check your roots by substituting them back into the original equation: ax² + bx + c should equal zero (within rounding error) for each root.
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