Quadratic Formula Calculator

Any equation you can write as ax² + bx + c = 0 can be solved with the quadratic formula. Enter the three coefficients and this calculator gives the solutions, real or complex, tells you why there are two, one or none that are real, and finds the parabola’s vertex.

How to Use It

1. Rearrange your equation so one side is 0, in the form ax² + bx + c = 0.
2. Enter a, b and c, keeping each sign. Use 0 for a missing term.
3. Press Calculate to see the solutions, the discriminant, the vertex and the axis of symmetry.

How It Works

The quadratic formula gives both solutions at once:

The part under the square root, the discriminant $b^2 - 4ac$, decides what kind of solutions you get: two real ones if it’s positive, one repeated one if it’s 0, and a pair of complex ones if it’s negative.

The vertex of the parabola $y = ax^2 + bx + c$ sits at $x = -\frac{b}{2a}$; put that x back into the equation to get its y. The vertical line through it is the axis of symmetry.

Worked Example

Solve $x^2 - 5x + 6 = 0$, so a = 1, b = −5 and c = 6:

The vertex is at $x = \frac{5}{2} = 2.5$, where $y = 2.5^2 - 5(2.5) + 6 = -0.25$.

Tips and Common Mistakes

• Set it equal to 0 first. For $x^2 = 5x - 6$, move everything to one side: $x^2 - 5x + 6 = 0$.
• Keep the signs. In $x^2 - 5x + 6$, b is −5, not 5; a sign slip changes both answers.
• Square all of b. $(-5)^2 = 25$, so a negative b still gives a positive $b^2$.
• Divide the whole top by 2a, not just the square root.
• Check your answers. Put each solution back into the equation; it should give 0.