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.

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 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.