Quadratic Formula Calculator
Solve ax² + bx + c = 0
Solving Quadratic Equations
Every quadratic equation of the form ax² + bx + c = 0 can be solved directly with a single formula — no factoring or guesswork required, even when the roots are complex numbers.
The value under the square root, b² − 4ac, instantly tells you how many real solutions exist and whether the parabola crosses, touches, or misses the x-axis entirely.
The vertex is the highest or lowest point on the parabola. It's a critical value in optimization problems — from maximizing area to minimizing cost — anywhere a quantity follows a quadratic relationship.
The Quadratic Formula
The vertex coordinates follow directly from the same coefficients: \( h = -\dfrac{b}{2a} \) and \( k = c - \dfrac{b^2}{4a} \), with the axis of symmetry being the vertical line \( x = h \).
Interpreting the Roots
When the discriminant is positive, the parabola crosses the x-axis at two distinct points. When it's exactly zero, the parabola is tangent to the x-axis at a single repeated root — the vertex itself sits on the axis. When it's negative, the parabola never touches the x-axis, and the two solutions become complex conjugates of the form p ± qi.
A positive discriminant means the graph crosses the x-axis at two separate points.
A zero discriminant means the vertex sits exactly on the x-axis.
A negative discriminant means the graph never touches the x-axis at all.
Key Takeaways
- The discriminant predicts the root type before you finish solving — positive, zero, or negative maps directly to two real, one real, or two complex roots.
- The vertex is the parabola's turning point, essential for any optimization problem modeled by a quadratic relationship.
- "a" must never be zero, since that would eliminate the x² term and reduce the equation to a linear one.
Frequently Asked Questions
- Enter the coefficients
a,b, andcfrom your equation in the formax² + bx + c = 0. - Click "Calculate" to see the discriminant, the type of roots, the root value(s), the vertex, and the axis of symmetry.
The discriminant is the expression b² − 4ac found under the square root in the quadratic formula. Its sign tells you what kind of roots the equation has before you even solve it: positive means two distinct real roots, zero means exactly one repeated real root, and negative means two complex (non-real) roots.
A negative discriminant means the parabola never crosses the x-axis, so there are no real solutions. Instead, the two roots are complex conjugates of the form p ± qi, where p is the real part and qi is the imaginary part. The calculator computes and displays both automatically.
Every parabola y = ax² + bx + c has a single turning point called the vertex, at coordinates (h, k) where h = −b / (2a) and k is the value of the function at that point. The axis of symmetry is the vertical line x = h that passes through the vertex, splitting the parabola into two mirror-image halves.
If a = 0, the x² term disappears and the equation becomes linear (bx + c = 0) rather than quadratic. The quadratic formula also divides by 2a, which is undefined when a is zero, so the calculator blocks this input with a clear error.
Yes, as long as you first rearrange your equation into standard form ax² + bx + c = 0 by moving every term to one side. Once it's in that form, just read off the coefficients (including their signs) and enter them directly.