Quadratic formula guide

How to use the Quadratic Formula Calculator

The Quadratic Formula Calculator solves equations in standard form ax^2 + bx + c = 0. It shows real or complex roots, the discriminant, the vertex, the axis of symmetry, and the steps used to reach the answer.

Open the quadratic calculator

Quick start

Rewrite your equation in standard form: ax^2 + bx + c = 0.
Enter the coefficient a from the x^2 term.
Enter the coefficient b from the x term.
Enter the constant c.
Press Calculate roots and review the roots, discriminant, vertex, axis, and steps.

The quadratic formula

The calculator uses x = (-b +/- sqrt(b^2 - 4ac)) / 2a. The expression inside the square root, b^2 - 4ac, is called the discriminant.

For x^2 - 3x + 2 = 0, the coefficients are a = 1, b = -3, and c = 2. The roots are x = 2 and x = 1.

Reading the discriminant

Positive The equation has two real roots.

Zero The equation has one repeated real root.

Negative The equation has two complex conjugate roots.

Examples from the quadratic calculator

Two real roots x^2 - 3x + 2 = 0

x = 2, 1

Repeated root x^2 - 4x + 4 = 0

x = 2

Complex roots x^2 + 2x + 5 = 0

x = -1 +/- 2i

Graph details

The calculator also shows the vertex and axis of symmetry. The axis of symmetry is x = -b / 2a, and the vertex is the point where the parabola reaches its maximum or minimum value.

If a is positive, the parabola opens up. If a is negative, it opens down. The constant c is the y-intercept.

Common mistakes

Make sure the equation is equal to zero before entering coefficients. For example, x^2 = 3x - 2 should be rewritten as x^2 - 3x + 2 = 0.

Coefficient a cannot be zero. If a is zero, the equation is linear, not quadratic, and the quadratic formula does not apply.

History, privacy, and copying

Recent quadratic answers stay visible in the current browser tab while you work. They are not sent to a server.

Copy answer copies the equation and roots so you can paste the result into notes, homework, a message, or another document.