Guide
Quadratic Formula Calculator Guide
Use this guide to understand how the quadratic formula works, what the discriminant means, and how roots and vertex details describe a parabola.
What is the Quadratic Formula?
The quadratic formula solves equations written in standard form as ax2 + bx + c = 0. It is useful when factoring is difficult or when the roots are decimal or complex values.
x = (-b +/- sqrt(b^2 - 4ac)) / 2a The plus-minus part creates two roots when the discriminant is positive. If the discriminant is zero, both roots are the same.
Understanding the Discriminant
The discriminant is the expression under the square root: b2 - 4ac. It tells you whether a quadratic equation has two real roots, one repeated real root, or two complex roots.
Positive discriminant
The equation has two real roots and the parabola crosses the x-axis twice.
Zero discriminant
The equation has one repeated real root and the parabola touches the x-axis once.
Negative discriminant
The equation has two complex roots and the parabola does not cross the x-axis.
Example Calculation
For the equation x^2 - 5x + 6 = 0, the coefficients are a = 1, b = -5, and c = 6. The discriminant is positive, so the equation has two real roots.
Example roots
x = 3 and x = 2 These are the x-values where the equation equals zero.How to Use This Calculator
- 1Write the equation in standard form
Move all terms to one side so it reads ax2 + bx + c = 0.
- 2Enter the value of a
Use the coefficient of x2. It cannot be 0.
- 3Enter b and c
Add the x coefficient and the constant term, including negative signs.
- 4Click Calculate
Review roots, discriminant meaning, vertex, and axis of symmetry.
- 5Check the root type
Use the discriminant result to know whether the solutions are real or complex.
Standard Form vs. Vertex Form
Standard form is ax2 + bx + c = 0. It is the easiest form for using the quadratic formula because the coefficients a, b, and c are visible.
Vertex form is a(x - h)2 + k. It highlights the vertex of the parabola at (h, k). This calculator starts from standard form, then calculates the vertex using x = -b / 2a and y = f(x).