Quadratic Equation Calculator

What a Quadratic Equation Is

A quadratic equation ($ax^2 + bx + c = 0$) is what you get whenever a squared term ($x^2$) is in the mix - the math behind a thrown ball’s path or the area of a growing square. Its graph is always a smooth U-shaped curve called a parabola, and “solving” it means finding where that curve crosses the x-axis (the spots where $y = 0$).

The graph under the calculator draws your parabola and marks those crossings. A curve can cross the axis twice (two real answers), just touch it once (one repeated answer), or miss it entirely (no real answer - the roots are complex). One number, the discriminant $b^2 - 4ac$, tells you which case you are in before you even solve.

The Quadratic Formula

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

The discriminant $D = b^2 - 4ac$ decides the type of roots:

• $D > 0$ — two real roots
• $D = 0$ — one repeated root
• $D < 0$ — two complex roots

How to Use It

1. Enter the coefficients a, b, and c.
2. Read the roots and the discriminant in the steps.

Worked Examples

a	b	c	Roots
1	−5	6	$x = 3$ or $x = 2$
1	−4	4	$x = 2$ (double root)
1	2	5	$x = -1 \pm 2i$ (complex)
2	1	−6	$x = 1.5$ or $x = -2$

FAQ

What is the quadratic formula?

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. It solves any quadratic once written as $ax^2 + bx + c = 0$.

What does the discriminant tell you?

The discriminant $b^2 - 4ac$ shows the type of roots: positive gives two real roots, zero gives one repeated root, and negative gives two complex roots.

What if a is 0?

Then it is not quadratic but linear ($bx + c = 0$), and the tool solves it as $x = -c/b$.