Quadratic Formula Calculator
Solve quadratic equations ax^2+bx+c=0 using the quadratic formula x=(-b+/-sqrt(b^2-4ac))/2a. Get real and complex roots, discriminant, vertex, and interactive graph.
About This Calculator
The Quadratic Formula Calculator solves any quadratic equation of the form ax^2 + bx + c = 0 using the quadratic formula x = (-b +/- sqrt(b^2 - 4ac)) / 2a. This calculator is designed for students, teachers, and professionals who need to quickly find roots, the discriminant, and the vertex of a parabola. Simply enter the three coefficients a, b, and c, and the calculator instantly computes the solutions along with key mathematical properties.
The discriminant Delta = b^2 - 4ac determines the nature of the solutions: Delta > 0 gives two distinct real roots, Delta = 0 gives one repeated real root, and Delta < 0 gives two complex conjugate roots. The vertex of the parabola y = ax^2 + bx + c is located at x = -b/(2a), which is the axis of symmetry. The y-coordinate of the vertex is found by substituting this x back into the original equation. For quadratic equations with real roots, the sum of the roots equals -b/a and the product of roots equals c/a.
This tool also generates an interactive parabola graph showing the curve and its key features. The graph plots the quadratic function over a range that centers on the vertex so you can visualize the shape and roots clearly. For additional analysis, a Key Metrics bar chart displays the discriminant, vertex coordinates, sum of roots, and product of roots. The calculator supports shareable permalinks: include the coefficients a, b, and c as URL parameters to share specific problems or save your work.
Quadratic equations appear across many fields including physics (projectile motion, kinematics), engineering (structural analysis, optimization), economics (profit maximization, cost analysis), and computer graphics (curve rendering, collision detection). Whether you are studying algebra, preparing for exams, or solving real-world problems, this tool provides instant solutions with visual context.
Frequently Asked Questions
What is the quadratic formula?
x = (-b +/- sqrt(b^2 - 4ac)) / 2a, where a, b, c are coefficients of the quadratic equation ax^2 + bx + c = 0.
What does the discriminant tell you?
The discriminant Delta = b^2 - 4ac determines the nature of roots: Delta > 0 means two distinct real roots, Delta = 0 means one repeated real root, and Delta < 0 means two complex conjugate roots.
What is the vertex of a parabola?
The vertex is the turning point of the parabola y = ax^2 + bx + c. Its x-coordinate is -b/(2a) and the y-coordinate is found by substituting this x back into the equation.
How do you handle complex roots?
When the discriminant Delta is negative, the quadratic has two complex conjugate roots of the form p +/- qi, where p = -b/(2a) is the real part and q = sqrt(-Delta)/(2a) is the imaginary part.
Can a quadratic have only one root?
Yes, when the discriminant Delta = 0, the quadratic has exactly one repeated real root (a double root). This occurs when the parabola touches the x-axis at exactly one point.
What if coefficient a is zero?
If a = 0, the equation bx + c = 0 is linear, not quadratic. The quadratic formula requires a ≠ 0 to work properly.
How do you graph a quadratic function?
Plot the vertex, find the y-intercept (x=0) and x-intercepts (roots), then use symmetry about the vertical line x = -b/(2a) to sketch the parabola.
What is the relationship between roots and coefficients?
For ax^2 + bx + c = 0, the sum of roots = -b/a and the product of roots = c/a. These relationships hold for both real and complex roots.