Solve any quadratic equation. This free quadratic formula calculator finds the roots (real or complex) of ax² + bx + c = 0, plus the discriminant, vertex, axis of symmetry and sum/product of roots.
ax² + bx + c = 0
x²
x
Enter the coefficients a, b and c. The calculator solves the equation with the quadratic formula and shows real or complex roots, plus the discriminant and vertex. (a can’t be 0.)
Roots (x)
—
Enter a, b and c
Discriminant
—
Nature of roots
—
Vertex
—
Axis of symmetry
—
Sum of roots
—
Product of roots
—
Uses the quadratic formula x = (−b ± √(b²−4ac)) ÷ 2a. When the discriminant is negative, the two roots are complex conjugates.
Use this free quadratic formula calculator to solve any quadratic equation ax² + bx + c = 0. Enter the coefficients a, b and c to get the roots, the discriminant and the vertex — including complex roots.
What this quadratic formula calculator does
A quadratic formula calculator solves equations of the form ax² + bx + c = 0. It finds the roots (real or complex), the discriminant, the vertex and axis of symmetry, and the sum and product of the roots.
The quadratic formula
x = ( −b ± √(b² − 4ac) ) ÷ 2a
Discriminant D = b² − 4ac
For x² − 5x + 6 = 0, the discriminant is 1 and the roots are x = 3 and x = 2.
What the discriminant tells you
Discriminant
Roots
D > 0
Two distinct real roots
D = 0
One repeated real root
D < 0
Two complex conjugate roots
Finding the vertex
The vertex of the parabola is at x = −b ÷ 2a, with the y-value found by substituting back in. It’s also the axis of symmetry. For x² − 5x + 6, the vertex is at (2.5, −0.25).
How to use the quadratic formula calculator
Enter coefficient a (it can’t be zero).
Enter b and c.
Read the roots, discriminant, vertex and more.
Quadratic Formula FAQ
What is the quadratic formula?
It solves ax² + bx + c = 0: x = (−b ± √(b² − 4ac)) ÷ 2a. The ± gives the two roots, and the part under the root, b² − 4ac, is the discriminant.
How do I solve a quadratic equation?
Identify a, b and c, then plug them into the quadratic formula. This calculator does the arithmetic and also handles complex roots automatically.
What does the discriminant tell me?
If it's positive there are two real roots, if zero there's one repeated real root, and if negative there are two complex conjugate roots.
What if a equals zero?
Then the equation isn't quadratic — it's linear (bx + c = 0). The calculator requires a non-zero a, since dividing by 2a wouldn't be defined.
How do I find the vertex?
The vertex is at x = −b ÷ 2a, with the y-value found by substituting that x back into the equation. It also marks the axis of symmetry.
Is the quadratic formula calculator free?
Yes, this quadratic formula calculator is completely free, needs no sign-up, and gives instant results in your browser.
