Quadratic Formula Calculator

Our quadratic equation calculator is here to find solutions (roots) and check your work—but it does not provide any shortcuts. Even though our calculator is efficient at finding an answer to your problems, it doesn’t reveal any of the steps involved in solving a quadratic equation. To better learn how to solve the equations on your own—and because practice helps improve your skills—we encourage you to solve the problems on your own first and use our calculator to make sure your answer is correct last. 

How to Use the Quadratic Equation Calculator

To use the calculator:

  1. Enter the corresponding values into the boxes below and click Solve.
  2. The results will appear in the boxes labeled Root 1 and Root 2. For example, for the quadratic equation below, you would enter 1, 5 and 6.
  3. After pressing Solve, your resulting roots would be -2 and -3.
  4. Click Reset to clear the calculator and enter new values.

x2 + 5x + 6 = 0

Enter the coefficient of x2 here
Enter the coefficient of x here
Enter the constant here

Root 1: 
Root 2: 

Important Terms for Quadratic Equations

Important Terms for Quadratic Equations

In case you’re new, need a refresher, or appreciate the knowledge, here are some helpful terms and descriptions to assist your calculations.

Quadratic Equations

A quadratic equation (also referred to as a quadratic function) is a polynomial whose highest exponent is 2. The standard form of a quadratic equation looks like this:

f (x) = ax² + bx + c

When graphed on a coordinate plane, a quadratic equation creates a parabola, which is a u-shaped curve. When the leading coefficient is positive, the curve is oriented like the letter u, with the opening facing up. When the leading coefficient is negative, the curve is upside down, with the opening facing down.


The coefficient of x² is called the leading coefficient, and is represented by the variable a. In standard form, a, b, and c are all constants or numerical coefficients. One absolute rule is that the first constant, a, can never be equal to zero.

The leading coefficient can tell you more than just the orientation of the parabola, it also determines how wide or skinny the u-curve is. This depends on the value of the leading coefficient. The closer to zero the value is, the wider the curve will be. The farther away from 0 the number is, the skinnier the curve will be.

Structure of a Graph

The vertex of a parabola is the point at the bottom of the u curve. If you draw a vertical line through the vertex, you create the axis of symmetry, which is an imaginary line that cuts the parabola in half equally. The shape of the curve is reflected over this line.

Quadratic Formula

The quadratic formula is used to find the solution to a quadratic equation. The quadratic formula looks like this:

For ax2  + bx + c = 0 where a ≠ 0:

x= -b + √b2-4ac / 2a

The Roots

Every quadratic equation gives two values of the unknown variable (x) and these values are called roots of the equation. When you are asked to solve a quadratic equation, you are really being asked to find the roots (or solutions).

The roots of a quadratic function are the x-intercepts, which are the points where the parabola crosses the x-axis. The y-coordinate of points lying on the x-axis is zero. Therefore, to find the roots of a quadratic function, we make f(x) = 0, and solve the equation.

A quadratic equation has two roots which may be unequal real numbers, equal real numbers, or numbers which are not real. If a quadratic equation has two real equal roots, we say the equation has only one real solution. This occurs when the vertex is the parabola is the point that touches the x-axis.


The discriminant of a quadratic formula tells you about the nature of roots the equation has.

For example:

  • b2−4ac = 0, one real solution
  • b2−4ac > 0, two real solutions
  • b2−4ac < 0, two imaginary solutions

If the discriminant is a perfect square, the roots are rational and when it is not a perfect square, the roots are irrational.

Other Calculators