Quadratic Equation Solver
Solve any quadratic equation ax² + bx + c = 0. Find roots, discriminant, nature of roots and vertex.
What is a Quadratic Equation?
A quadratic equation is a second-degree polynomial equation of the form ax² + bx + c = 0, where a, b and c are constants and a is not equal to zero. Quadratic equations appear throughout mathematics, physics, engineering, economics and many other fields. The solutions (roots) of a quadratic equation are the values of x that make the equation true.
How to Use This Solver
Enter the values of coefficients a, b and c from your quadratic equation. The solver will calculate the discriminant, determine the nature of the roots (real and distinct, real and equal, or complex), compute both roots, and find the vertex of the parabola. The equation is displayed in formatted form as you enter the coefficients.
Formula
The quadratic formula is:
x = (-b ± √(b² - 4ac)) / 2a
The discriminant D = b² - 4ac determines the nature of the roots:
D > 0: Two real and distinct roots
D = 0: Two real and equal roots (one repeated root)
D < 0: Two complex conjugate roots
The vertex of the parabola is at: (-b/2a, f(-b/2a))
Frequently Asked Questions
What happens if a is zero?
If a = 0, the equation is not quadratic but linear (bx + c = 0). This solver requires a to be non-zero. For a linear equation, the solution is simply x = -c/b.
What are complex roots?
When the discriminant is negative, the square root of a negative number is involved, leading to complex (imaginary) roots. These are expressed in the form a + bi and a - bi, where i is the imaginary unit (√-1).
What is the vertex of a parabola?
The vertex is the highest or lowest point of the parabola defined by the quadratic equation. If a > 0, the parabola opens upward and the vertex is the minimum point. If a < 0, it opens downward and the vertex is the maximum point.
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