Quadratic Equation Calculator

A quadratic equation has the form ax² + bx + c = 0. The quadratic formula gives all solutions. The discriminant (b² − 4ac) tells you what kind of roots to expect before you solve: two real roots, one repeated root, or two complex roots.

• Algebra homework: Solve quadratic equations and check factored or completed-square solutions.
• Physics projectile motion: The height of a thrown object over time is a quadratic function. Find when it hits the ground (h = 0).
• Engineering design: Optimize parabolic structures or find equilibrium points in systems with quadratic relationships.
• Business break-even: Profit functions are sometimes quadratic. Find zero-profit points using this solver.

1. 1Enter the three coefficients a, b, and c for your quadratic equation ax² + bx + c = 0.
2. 2The calculator computes the discriminant D = b² − 4ac.
3. 3If D > 0, there are two distinct real roots.
4. 4If D = 0, there is one repeated root (the parabola touches the x-axis at one point).
5. 5If D < 0, there are two complex (imaginary) roots. The parabola does not intersect the x-axis.
6. 6The vertex of the parabola is also shown.

• Entering coefficients in the wrong order: The three inputs correspond to ax² + bx + c = 0. "a" is the coefficient of x², "b" is the coefficient of x, and "c" is the constant. Entering them out of order gives wrong roots.
• Forgetting the sign of negative coefficients: If the equation is x² - 5x + 6 = 0, enter b = -5, not 5. Missing the negative sign produces a completely different pair of roots.
• Entering a = 0: If the leading coefficient is zero, the equation is linear, not quadratic. The quadratic formula requires a to be non-zero. Use a linear equation solver or simple algebra for those cases.
• Expecting real roots when the discriminant is negative: A negative discriminant (b² - 4ac < 0) means no real roots exist. The roots are complex numbers. The calculator shows this, but it means there is no real-world x where the parabola crosses zero.

• Check roots by substituting back: After finding the roots, substitute each back into the original equation. Both should equal zero (or very close to it, accounting for floating-point rounding). This confirms the calculation is correct.
• Use the discriminant to classify the parabola: Discriminant > 0: two distinct real roots, parabola crosses the x-axis twice. Discriminant = 0: one repeated root, parabola touches the axis at one point. Discriminant < 0: no real roots, parabola does not cross.
• Rearrange the equation before entering coefficients: If your equation is in a different form (e.g., 3x² = 12 - x), rearrange to ax² + bx + c = 0 first: 3x² + x - 12 = 0. Then enter a=3, b=1, c=-12.
• Use for projectile motion problems: Time of flight in projectile problems often involves a quadratic: -4.9t² + v₀t + h₀ = 0. Enter the coefficients directly and take the positive root as the time of landing.