Quadratic Equation Solver
Solve any quadratic equation ax² + bx + c = 0 for its roots, discriminant, and vertex form.
ax² + bx + c = 0
Discriminant
1
Roots
x₁ = 2, x₂ = 1
Vertex
(1.5, -0.25)
How to Solve a Quadratic Equation
A quadratic equation has the form ax² + bx + c = 0, where a, b, and c are coefficients and a ≠ 0. Enter the three coefficients below to find its roots using the quadratic formula, along with the discriminant and the equation's vertex form.
The discriminant (b² − 4ac) tells you what kind of roots to expect: positive means two distinct real roots, zero means one repeated real root, and negative means two complex (imaginary) roots.
Example
For x² − 3x + 2 = 0 (a=1, b=−3, c=2), the discriminant is (−3)² − 4(1)(2) = 1, which is positive, giving two real roots: x = 2 and x = 1.
Common Use Cases
- Solving algebra homework problems involving quadratic equations.
- Finding the roots of a parabola, such as where a projectile's height equals zero.
- Finding a parabola's vertex (maximum or minimum point) for graphing or optimization problems.
FAQs
What does a negative discriminant mean?
A negative discriminant means the equation has no real roots — the parabola never crosses the x-axis. Instead, it has two complex conjugate roots of the form p ± qi, where i is the imaginary unit (√−1).
What is vertex form and why does it matter?
Vertex form rewrites the equation as a(x − h)² + k, where (h, k) is the vertex — the parabola's highest or lowest point. It's useful for quickly reading off the maximum or minimum value of the expression without solving for the roots.
What if I enter a = 0?
With a = 0 the equation is no longer quadratic — it becomes linear (bx + c = 0) — so this calculator requires a non-zero value for a to solve it as a proper quadratic equation.