Quadratic Formula Calculator
Solve any quadratic equation ax² + bx + c = 0 for its real or complex roots, with the discriminant, vertex, and a plot of the parabola.
Results from this calculator are estimates provided for general informational purposes only, based on formulas, rates, and standards commonly accepted as of 2026. Figures may differ slightly from other calculators or professional sources due to rounding methods, differing assumptions, or regional regulations, and rules may change over time. Always consult a qualified professional — such as a financial advisor, healthcare provider, or other relevant specialist — before making decisions based on these results.
Roots
x₁ = 2, x₂ = 1
For x² - 3x + 2 = 0, the discriminant is 1, giving two real roots: x = 2 and x = 1.
Discriminant
1
Vertex
(1.5, -0.25)
Parabola Opens
Upward
Real Roots?
Yes
What Is a Quadratic Equation?
A quadratic equation is any second-degree polynomial equation of the form ax² + bx + c = 0, where a cannot equal zero (otherwise it reduces to a linear equation). Its solutions — called roots — are the x-values where the corresponding parabola y = ax² + bx + c crosses the x-axis.
Quadratic equations show up constantly in real-world modeling: projectile motion, area optimization problems, and any relationship where a quantity depends on the square of another variable.
Plot of y = ax² + bx + c
The Quadratic Formula
Any quadratic equation can be solved directly using the quadratic formula, derived by completing the square on the general form ax² + bx + c = 0.
The Discriminant Determines the Type of Roots
The expression under the square root, b² − 4ac, is called the discriminant. When it's positive, the equation has two distinct real roots. When it equals zero, there's exactly one repeated real root (the parabola's vertex touches the x-axis). When it's negative, the roots are a complex-conjugate pair, and the parabola never crosses the x-axis at all.
The Vertex and Axis of Symmetry
Every parabola is symmetric around a vertical line called the axis of symmetry, located at x = −b ÷ 2a — which is also the x-coordinate of the vertex, the parabola's minimum point (if a > 0) or maximum point (if a < 0).
Real-World Applications
Projectile motion under gravity follows a quadratic path, so the quadratic formula finds exactly when a thrown object hits the ground. Quadratics also solve optimization problems like maximizing a rectangular area for a fixed perimeter, and appear throughout physics, engineering, and finance wherever a squared relationship applies.
Example — Your Current Inputs
For x² - 3x + 2 = 0, the discriminant is 1, giving two real roots: x = 2 and x = 1.
Additional Example — A Ball Thrown Upward
A ball thrown upward follows height h = −16t² + 64t + 5 (feet, seconds). Setting h = 0 and solving −16t² + 64t + 5 = 0 with the quadratic formula gives t ≈ 4.08 seconds — the moment the ball hits the ground, ignoring the negative time root, which isn't physically meaningful.
About These Parameters
- Coefficient a
- The coefficient of x², controlling whether the parabola opens upward (a > 0) or downward (a < 0), and how narrow or wide it is. Cannot be zero.
- Coefficient b
- The coefficient of the linear x term, which shifts the parabola's vertex left or right and affects the axis of symmetry.
- Coefficient c
- The constant term — also the parabola's y-intercept, the value of y when x = 0.
Frequently Asked Questions
What does it mean when the roots are complex?
A negative discriminant means the parabola never crosses the x-axis — there's no real number solution. The roots still exist mathematically as a complex-conjugate pair, useful in fields like electrical engineering, but they don't correspond to points on a standard x-y graph.
Can I use fractions as coefficients?
Yes — enter a decimal equivalent (like 0.75 for 3/4) in any of the coefficient fields, and the calculator solves the equation exactly the same way.
What if a is zero?
If a = 0, the x² term disappears and the equation becomes linear (bx + c = 0), which has at most one solution and isn't a quadratic equation anymore — this calculator requires a nonzero value for a.