Quadratic Formula Calculator

Enter the coefficients, $\hspace{0.2em} a \hspace{0.2em}$ , $\hspace{0.2em} b \hspace{0.2em}$ , and $\hspace{0.2em} c \hspace{0.2em}$

\hspace{0.2em} x^2\,+ \hspace{0.2em}

\hspace{0.2em} x\,+ \hspace{0.2em}

\hspace{0.2em} = 0 \hspace{0.2em}

Please provide your input and click the calculate button

About the Quadratic Formula Calculator

This quadratic formula calculator lets you calculate the discriminant and the roots or solutions for a quadratic equation.

The calculator will tell you not only the answers but also how to find the discriminant and solve the quadratic equation using the quadratic formula.

Usage Guide

Hide

i. Valid Inputs

Each of the three inputs can be any real number (with one exception, mentioned below). Here are a few examples.

Whole numbers or decimals → $\hspace{0.2em} 2 \hspace{0.2em}$ , $\hspace{0.2em} -4.25 \hspace{0.2em}$ , $\hspace{0.2em} 0 \hspace{0.2em}$ , $\hspace{0.2em} 0.33 \hspace{0.2em}$
Fractions → $\hspace{0.2em} 2/3 \hspace{0.2em}$ , $\hspace{0.2em} -1/5 \hspace{0.2em}$
Mixed numbers → $\hspace{0.2em} 5 \hspace{0.5em} 1/4 \hspace{0.2em}$

NOTE — The first input, the coefficient of $\hspace{0.2em} x^2 \hspace{0.2em}$ , must be a non-zero real number. (If $\hspace{0.2em} a = 0 \hspace{0.2em}$ , the second-degree term would vanish and it won't be a quadratic equation.)

ii. Example

If you would like to see an example of the calculator's working, just click the "example" button.

iii. Solutions

As mentioned earlier, the calculator won't just tell you the answer but also the steps you can follow to do the calculation yourself. The "show/hide solution" button would be available to you after the calculator has processed your input.

iv. Share

We would love to see you share our calculators with your family, friends, or anyone else who might find it useful.

By checking the "include calculation" checkbox, you can share your calculation as well.

Here's a quick overview of what the quadratic formula is and how to use it.

For those interested, we have a more comprehensive tutorial on the quadratic formula.

The Quadratic Formula

The quadratic formula allows us to solve quadratic equations regardless of the nature of its roots. Here's how.

Consider the following quadratic equation.

ax^2 + bx + c = 0

According to the quadratic formula, the solutions $\hspace{0.2em} (x_1, x_2) \hspace{0.2em}$ to this quadratic equation would be —

x_1, x_2 \hspace{0.25em} = \hspace{0.25em} \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Let's lok at an example.

Example

Solve the quadratic equation

\hspace{0.2em} x^2 + 3x - 4 = 0 \hspace{0.2em}

using the quadratic formula.

Solution

Let's start by identifying the coefficients $\hspace{0.2em} a \hspace{0.2em}$ , $\hspace{0.2em} b \hspace{0.2em}$ , and $\hspace{0.2em} c \hspace{0.2em}$ .

We do that by comparing the given equation with the standard equation. So

a \hspace{0.2em} = \hspace{0.2em} 1, \hspace{1em} b \hspace{0.2em} = \hspace{0.2em} 3, \hspace{1em} c \hspace{0.2em} = \hspace{0.2em} -4

Substituting these values into the quadratic formula.

\begin{align*} x_1, x_2 \hspace{0.25em} &= \hspace{0.25em} \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \\[1.75em] &= \hspace{0.25em} \frac{-3 \pm \sqrt{3^2 - 4 \cdot 1 \cdot (-4)}}{2 \cdot 1} \\[1.75em] &= \hspace{0.25em} \frac{-3 \pm 5}{2} \end{align*}

Separating the two values, we get —

x_1 \hspace{0.2em} = \hspace{0.2em} \frac{-3 + 5}{2}, \hspace{1em} x_2 \hspace{0.2em} = \hspace{0.2em} \frac{-3 - 5}{2}

And so,

x_1 \hspace{0.2em} = \hspace{0.2em} 1, \hspace{1em} x_2 \hspace{0.2em} = \hspace{0.2em} -4

Equation solved.

The Discriminant

In the quadratic formula, the part under the radical symbol is known as the discriminant. It's often denoted by the symbol $\hspace{0.2em} \Delta \hspace{0.2em}$ . So,

\Delta \hspace{0.25em} = \hspace{0.25em} b^2 - 4ac

The discriminant lets us detemine the nature of the equation's roots without having to solve it. All we need to do is evaluate the discriminant.

TK - discriminant table

Here's an example.

Example

Without actually solving the quadratic equation

\hspace{0.2em} 4x^2 - 12x + 9 = 0 \hspace{0.2em}

, determine the nature of its roots.

Solution

As I just mentioned, to determine the nature of its roots, we just need to evaluate the discriminant. So, let's do that.

\Delta \hspace{0.25em} = \hspace{0.25em} b^2 - 4ac

Substituting the values of $\hspace{0.2em} a \hspace{0.2em}$ , $\hspace{0.2em} b \hspace{0.2em}$ , and $\hspace{0.2em} c \hspace{0.2em}$ —

\begin{align*} \Delta \hspace{0.25em} &= \hspace{0.25em} (-12)^2 - 4 \cdot 4 \cdot 9 \\[1em] &= \hspace{0.25em} 0 \end{align*}

Now from the table above, we can see that a discriminant of $\hspace{0.2em} 0 \hspace{0.2em}$ means the roots of the equation are real and equal.