Math
Quadratic Equation Calculator
Solve ax² + bx + c = 0 instantly. Finds real and complex roots, discriminant, and vertex coordinates.
ax² + bx + c = 0
Enter values for a, b, and c above (a ≠ 0)
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The Quadratic Formula
x = (−b ± √(b² − 4ac)) / 2a
Where a, b, and c are the coefficients of the equation ax² + bx + c = 0.
The discriminant Δ = b² − 4ac determines the nature of the roots.
The vertex x-coordinate = −b / 2a; y-coordinate = c − b² / 4a.
How to Use This Calculator
Enter the coefficients a, b, and c from your equation. The coefficient a cannot be zero.
The calculator will show the roots, the discriminant value, and the parabola vertex. Complex roots are displayed in a + bi form.
Frequently asked questions
What is the quadratic formula?
The quadratic formula is x = (−b ± √(b²−4ac)) / 2a. It solves any equation of the form ax² + bx + c = 0 for the values of x.
What is the discriminant and what does it tell you?
The discriminant is b² − 4ac. If it is positive, the equation has two distinct real roots. If it equals zero, there is one repeated real root. If it is negative, the roots are complex (imaginary).
When does a quadratic equation have complex roots?
When the discriminant (b² − 4ac) is negative, the square root of a negative number appears in the formula, producing two complex conjugate roots of the form p ± qi.
What is the vertex of a parabola?
The vertex is the highest or lowest point on the parabola. Its x-coordinate is −b / 2a and its y-coordinate is found by substituting that x value back into the equation.
Where are quadratic equations used in real life?
Quadratic equations model projectile motion, the profit-maximising output in economics, the shape of satellite dishes, and many problems in engineering and physics.