Quadratic Formula Calculator
Solve any quadratic equation with this quadratic formula calculator. Shows roots, discriminant, vertex, step-by-step work, and a parabola graph.
About Quadratic Formula Calculator
The quadratic formula is one of the most important tools in algebra. This calculator lets you enter the coefficients a, b, and c for any equation in the form ax2 + bx + c = 0 and instantly see the solution.
How It Works
Enter the three coefficients and the calculator applies the quadratic formula: x = (-b +/- sqrt(b2 - 4ac)) / 2a. It first computes the discriminant to determine whether the roots are real or complex, then calculates both roots and the vertex of the parabola.
Step-by-Step Breakdown
Every solution includes a numbered walkthrough so you can follow the math. Each step shows the substitution and arithmetic, making this a great study companion for algebra homework. The calculator also displays a simple parabola sketch with the roots and vertex marked, helping you visualise the equation.
Real and Complex Roots
The three discriminant cases are clearly labeled. Positive discriminant gives two real roots, zero gives one repeated root, and negative gives complex conjugate roots displayed in a + bi form. A color-coded badge shows which case applies at a glance.
Related Tools
For systems of equations or linear equations, try the Equation Solver. If you need to work with fractions along the way, the Fraction Calculator is a quick companion tool. All calculations run entirely in your browser with nothing sent to any server.
Frequently Asked Questions
What is the quadratic formula?
The quadratic formula is x = (-b +/- sqrt(b^2 - 4ac)) / 2a. It finds the solutions (roots) of any quadratic equation in the form ax^2 + bx + c = 0, where a is not zero.
What does the discriminant tell you?
The discriminant is b^2 - 4ac. If it is positive, the equation has two distinct real roots. If it equals zero, there is exactly one repeated root. If it is negative, the roots are complex (imaginary) numbers.
Can this calculator handle complex roots?
Yes. When the discriminant is negative, the calculator displays both complex conjugate roots in the form a + bi and a - bi, where i is the imaginary unit.
What is the vertex of a parabola?
The vertex is the highest or lowest point on the parabola, depending on whether it opens downward or upward. Its coordinates are (h, k) where h = -b/2a and k = f(h). This calculator computes the vertex automatically.
Why can't the coefficient a be zero?
If a equals zero, the equation becomes linear (bx + c = 0) rather than quadratic. A quadratic equation requires a nonzero x^2 term. Use the Equation Solver tool if you need to solve linear equations.