Quadratic Formula Calculator

What is a quadratic formula calculator?

A quadratic formula calculator solves a quadratic equation of the form $a x^2 + b x + c = 0$ for its real roots. You enter the three coefficients — the leading coefficient $a$, the linear coefficient $b$, and the constant term $c$ — and the calculator returns the discriminant together with the two real solutions $x_1$ and $x_2$, each rounded to four decimal places.

A quadratic equation is a second-degree polynomial equation, meaning the highest power of the unknown is two. As long as $a \neq 0$, the equation describes a parabola, and its real roots are exactly the points where that parabola crosses the horizontal axis.

How does it work?

The roots are found with the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

The expression under the square root, $b^2 - 4ac$, is called the discriminant and is usually written $\Delta$:

$\Delta = b^2 - 4ac$

The discriminant tells you how many real roots the equation has before you even compute them:

• If $\Delta > 0$, there are two distinct real roots.
• If $\Delta = 0$, there is one repeated real root (the two solutions coincide).
• If $\Delta < 0$, there are no real roots — the solutions are a complex-conjugate pair, so the calculator leaves the root fields empty.

The calculator also requires $a \neq 0$. When $a = 0$ the equation is no longer quadratic but linear, so no quadratic roots are reported.

Worked examples

Example 1 — two roots. Solve $x^2 - 3x + 2 = 0$, so $a = 1$, $b = -3$, $c = 2$.

$\Delta = (-3)^2 - 4 \cdot 1 \cdot 2 = 9 - 8 = 1$

$x = \frac{3 \pm \sqrt{1}}{2} = \frac{3 \pm 1}{2}$

This gives $x_1 = 2$ and $x_2 = 1$.

Example 2 — a repeated root. Solve $x^2 + 2x + 1 = 0$, so $a = 1$, $b = 2$, $c = 1$.

$\Delta = 2^2 - 4 \cdot 1 \cdot 1 = 4 - 4 = 0$

$x = \frac{-2 \pm \sqrt{0}}{2} = -1$

Both roots equal $-1$, the single point where the parabola touches the axis.

Example 3 — no real roots. Solve $x^2 + 1 = 0$, so $a = 1$, $b = 0$, $c = 1$.

$\Delta = 0^2 - 4 \cdot 1 \cdot 1 = -4$

Because $\Delta < 0$, there are no real solutions, so the calculator returns only the discriminant and leaves the root fields blank.

Practical notes

Sign matters: enter $b$ and $c$ exactly as they appear, including the minus sign, so type -3 for $b$ in the first example. Results are rounded to four decimal places, which is usually plenty for graphing, physics, and engineering work but means that irrational roots such as $\sqrt{2}$ are shown as their decimal approximation.

The quadratic formula is closely related to other algebra tools. Once you have the roots you can rebuild the equation in factored form $a(x - x_1)(x - x_2)$, which connects naturally to a factor calculator. The square-root step at the heart of the formula generalizes the idea behind a cube-root calculator, and the squared terms tie in with raising numbers to powers via an exponent calculator.