Free Online Quadratic Formula Calculator

Solve any quadratic equation ax² + bx + c = 0 instantly. Get real and complex roots with discriminant analysis and step-by-step solution.

Quadratic Equation Solver

Enter the coefficients a, b, and c for the equation ax² + bx + c = 0.

a (x² coefficient)

b (x coefficient)

c (constant)

Solutions

What is the Quadratic Formula?

The quadratic formula is a mathematical formula that provides the solution(s) to any quadratic equation of the form ax² + bx + c = 0, where a ≠ 0. It is one of the most important formulas in algebra, used by students, engineers, scientists, and economists worldwide.

x = (-b ± √(b² - 4ac)) / 2a

This formula was known to ancient Babylonian mathematicians as far back as 2000 BCE, though in a different form. The modern algebraic notation was developed during the Renaissance. Khan Academy's algebra course provides excellent video lessons on understanding and applying the quadratic formula.

Understanding the Discriminant

The discriminant (b² - 4ac) is the key to understanding the nature of a quadratic equation's solutions:

Discriminant (Δ) = b² - 4ac

Δ > 0: Two distinct real roots — the parabola crosses the x-axis at two points.
Δ = 0: One repeated real root (double root) — the parabola touches the x-axis at exactly one point (vertex).
Δ < 0: Two complex conjugate roots — the parabola doesn't cross the x-axis.

The discriminant concept extends to higher-degree polynomials and is fundamental in algebraic number theory.

Step-by-Step Example

Solve: 2x² - 7x + 3 = 0

Identify: a = 2, b = -7, c = 3

Discriminant: (-7)² - 4(2)(3) = 49 - 24 = 25

Square root: √25 = 5

Solutions:

x₁ = (7 + 5) / 4 = 12/4 = 3

x₂ = (7 - 5) / 4 = 2/4 = 0.5

Other Methods for Solving Quadratic Equations

Factoring

When a quadratic can be factored, it's often the fastest method. For example, x² - 5x + 6 = (x-2)(x-3) = 0 gives x = 2 or x = 3. Finding factors often relies on the Greatest Common Factor.

Completing the Square

This method transforms the equation into a perfect square trinomial. It's actually how the quadratic formula itself is derived, as explained in Purplemath's completing the square guide.

Graphing

The solutions (roots) of a quadratic equation are the x-intercepts of the corresponding parabola y = ax² + bx + c.

Real-World Applications

Physics — Projectile Motion: The trajectory of a thrown ball follows a parabolic path described by quadratic equations. NASA uses these equations for trajectory calculations.
Engineering: Bridge arches, satellite dishes, and car headlight reflectors are parabolic shapes designed using quadratic equations.
Business: Profit optimization models often involve quadratic relationships between price, quantity, and revenue. Try our Profit Margin Calculator for business calculations.
Architecture: Parabolic arches and domes use quadratic curves for structural stability, as documented by Architectural Digest.

Frequently Asked Questions

If a = 0, the equation becomes linear (bx + c = 0), not quadratic. The solution is simply x = -c/b. A quadratic equation must have a ≠ 0.

When the discriminant is negative, the square root of a negative number involves the imaginary unit i (where i² = -1). The roots come in conjugate pairs: a + bi and a - bi. Complex numbers are widely used in electrical engineering and quantum physics.

The quadratic formula is derived by completing the square on the general equation ax² + bx + c = 0. You isolate x by systematically transforming the equation into a perfect square on one side.

No, the quadratic formula only works for degree-2 equations. Cubic (degree 3) and quartic (degree 4) have their own formulas. Equations of degree 5 or higher have no general algebraic formula (proven by Abel-Ruffini theorem).