Quadratic Equation Solver
Solves ax² + bx + c = 0 using the quadratic formula. Works for both real and complex roots - so it'll give you the imaginary solutions too when the discriminant goes negative, not just tell you there are no solutions.
How to use
- Enter the three coefficients: a (for x²), b (for x), and c (the constant).
- Click Calculate. You'll see both roots, the discriminant, and whether the roots are real or complex.
Related tools:
The quadratic formula x = (-b ± sqrt(b^2 - 4ac)) / 2a solves any equation of the form ax^2 + bx + c = 0. The discriminant (b^2 - 4ac) determines the nature of the roots: positive means two distinct real roots, zero means one repeated real root, negative means two complex (imaginary) roots.
Quadratic equations appear in physics (projectile motion), economics (profit maximisation where revenue is quadratic in price), and geometry (areas, Pythagorean problems). The calculator shows both roots, the discriminant, and the vertex of the parabola so you have all the information in one place.
Frequently Asked Questions
What does the discriminant tell us?
Discriminant = b^2 - 4ac. If positive: two distinct real roots (parabola crosses x-axis twice). If zero: one repeated real root (parabola is tangent to x-axis). If negative: no real roots, only complex ones (parabola does not cross x-axis).
Can I solve a quadratic without the formula?
Often yes. Factoring works when the equation factors neatly: x^2 + 5x + 6 = (x+2)(x+3) = 0, giving roots -2 and -3. Completing the square is another method. The quadratic formula always works regardless of whether it factors nicely.
What are complex roots and when do they appear?
Complex roots involve the imaginary unit i (square root of -1). They appear when the discriminant is negative. In real-world physical problems, complex roots often mean there is no real solution to the scenario described (e.g. no time when a projectile reaches a given height if it was never that high).