Quadratic Formula Calculator

The quadratic formula is used to find the roots of a quadratic equation in the form

The solution is given by the equation

How to Use This Calculator

1. Enter values for coefficients a, b, and c.
2. Ensure a is nonzero.
3. View the computed roots and discriminant below.

Note: a must be nonzero because the variable a is in the denominator of the quadratic solution, and you cannot divide by zero.

Understanding the Discriminant

The discriminant is used to determine the nature of the roots. It's an equation derived from within the square root of the quadratic solution equation.

Based on the sign of the discriminant, the nature of the roots are revealed.

• If positive, the equation has two real roots.
• If zero, the equation has one real root (a repeated root).
• If negative, the equation has two complex roots.

Example Problems

Problem 1

Find the roots of the equation using the quadratic formula.

1. Extract the coefficients from the equation

   $a=5$$b=8$$c=-3$

2. Plug the coefficients into the equation and simplify

   $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$x=\frac{-8\pm\sqrt{8^2-4\left(5\right)\left(-3\right)}}{2\left(5\right)}$$x=\frac{-8\pm\sqrt{124}}{10}$$x=\frac{-8\pm11.136}{10}$

   Notice that the discriminant is negative, indicating that the quadratic has two complex roots.

   $x_1=0.314$$x_2=-1.914$$\text{discriminant}=124$

Problem 2

Find the roots of the equation using the quadratic formula.

1. Extract the coefficients from the equation

   $a=1$$b=5$$c=12$

2. Plug the coefficients into the equation and simplify

   $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$x=\frac{-5\pm\sqrt{5^2-4\left(1\right)\left(12\right)}}{2\left(1\right)}$$x=\frac{-5\pm\sqrt{-23}}{2}$$x=\frac{-5\pm4.796i}{2}$

   Notice that the discriminant is positive, indicating that the quadratic has two real roots.

   $x_1=-2.5+2.398i$$x_2=-2.5-2.398i$$\text{discriminant}=-23$