Quadratic Formula Calculator
Solve quadratic equations ax2 + bx + c = 0. Calculate real/complex roots, discriminant details, and parabolic vertex coordinate sets.
x₂ = 2
Parabolic Geometric Specifications
Continuous Parabolic Curve Representation
Step-by-step discriminant analysis solver
How the Quadratic Formula Solves Polynomials
A quadratic equation is a second-degree polynomial equation written in standard form:
ax² + bx + c = 0
where a ≠ 0. The roots represent the x-intercepts (where the curve crosses the horizontal axis). The general quadratic formula is solved by:
x = [-b ± √(b² - 4ac)] / 2a
Role of the Discriminant (D)
The expression inside the radical D = b² - 4ac is the discriminant. Its sign dictates the number and type of roots:
- D > 0: The curve intersects the x-axis twice, yielding two distinct real roots.
- D = 0: The curve touches the x-axis at a single vertex point, yielding one double/repeated real root.
- D < 0: The curve lies entirely above or below the x-axis, yielding two complex conjugate roots (imaginary coordinates containing i).
Frequently Asked Questions About Quadratic Equations
What is the quadratic formula?
The quadratic formula is a mathematical formula used to find the solutions (roots) of a quadratic equation ax² + bx + c = 0. The formula is x = (-b ± √(b² - 4ac)) / 2a.
What does a negative discriminant mean?
A negative discriminant (b² - 4ac < 0) indicates that the quadratic equation has no real roots. Instead, it has two complex conjugate roots containing the imaginary unit 'i'.
What are the vertex coordinates of a parabola?
The vertex coordinates (h, k) represent the maximum or minimum peak of the parabola. The horizontal coordinate is h = -b / (2a), and the vertical coordinate is k = f(h) = -(b² - 4ac) / (4a).
Can this solve quadratic equations with complex roots?
Yes. Get real and complex roots, discriminant analysis, and vertex details.