Quadratic Equation Calculator – Solve ax² + bx + c = 0
Solve quadratic equations and find roots using the quadratic formula
How to Use
- Enter coefficient a (cannot be zero)
- Enter coefficient b
- Enter coefficient c
- Click calculate to find the roots and properties
What is a Quadratic Equation?
A quadratic equation is a second-degree polynomial equation in the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The solutions to this equation are called roots or zeros.
The graph of a quadratic equation is a parabola, which opens upward if a > 0 and downward if a < 0.
The Quadratic Formula
The quadratic formula provides the solutions to any quadratic equation:
x = (-b ± √(b² - 4ac)) / (2a)
The expression under the square root, b² - 4ac, is called the discriminant (Δ) and determines the nature of the roots.
The Discriminant
The discriminant Δ = b² - 4ac tells us about the nature of the roots:
- If Δ > 0: Two distinct real roots
- If Δ = 0: One repeated real root (double root)
- If Δ < 0: Two complex conjugate roots
Vertex and Axis of Symmetry
The vertex of the parabola is at the point (-b/(2a), f(-b/(2a))), where f(x) = ax² + bx + c.
The axis of symmetry is the vertical line x = -b/(2a), which passes through the vertex.
Methods to Solve Quadratic Equations
- Quadratic Formula: Works for all quadratic equations
- Factoring: When the equation can be factored easily
- Completing the Square: Useful for deriving the quadratic formula
- Graphing: Finding x-intercepts of the parabola
Frequently Asked Questions
- Why can't coefficient 'a' be zero?
- If a = 0, the equation becomes bx + c = 0, which is a linear equation, not a quadratic equation. Quadratic equations must have an x² term.
- What are complex roots?
- Complex roots occur when the discriminant is negative. They involve the imaginary unit i = √(-1) and always come in conjugate pairs, like 2 + 3i and 2 - 3i.
- How do I know if my equation has real solutions?
- Calculate the discriminant Δ = b² - 4ac. If Δ ≥ 0, the equation has real solutions. If Δ < 0, the solutions are complex numbers.
- What is the relationship between roots and coefficients?
- For ax² + bx + c = 0 with roots r and s: the sum of roots r + s = -b/a, and the product of roots r × s = c/a. This is known as Vieta's formulas.