Solving any quadratic equation
A quadratic equation has the form a x squared plus b x plus c equals zero. This calculator finds its roots with the quadratic formula, tells you whether they are real or complex, and locates the parabola vertex and axis of symmetry — the full picture from three coefficients.
The quadratic formula
The roots are x = (-b plus or minus the square root of (b squared minus 4ac)) divided by 2a. The part under the root, b squared minus 4ac, is the discriminant, and its sign decides everything: positive gives two distinct real roots, zero gives one repeated root, and negative gives a conjugate pair of complex roots.
Vertex and axis
Every quadratic traces a parabola whose turning point sits at h = -b / 2a, with k found by substituting h back in. The vertical line x = h is the axis of symmetry, mirroring the two roots about the center.
Reading the discriminant
| Discriminant | Roots |
|---|---|
| Greater than 0 | Two distinct real |
| Equal to 0 | One repeated real |
| Less than 0 | Two complex conjugates |
Related algebra tools
For a straight line through two points, use the slope calculator; for two linear equations at once, the system of equations calculator.
Worked example
For x squared minus 5x plus 6 = 0, the discriminant is 25 minus 24 = 1, so the roots are real: x = 3 and x = 2, with the vertex at (2.5, -0.25).
FAQ
What if a is zero?
Then it is not quadratic but linear, and the single root is x = -c / b. The calculator detects this and switches to the linear solution.
What does a complex root look like here?
It is shown as a real part plus or minus an imaginary part times i, the two roots being mirror images across the real axis.