The Quadratic Formula, Explained

Quadratic equations show up everywhere — from projectile motion in physics to profit maximization in business. The good news: one formula solves them all.

The General Form

A quadratic equation has the shape:

where a, b and c are numbers, and a ≠ 0. The solutions are given by:

The Discriminant Decides Everything

The part under the square root, b² − 4ac, is called the discriminant. It tells you what kind of solutions to expect:

• Positive → two distinct real roots.
• Zero → exactly one repeated real root.
• Negative → two complex roots (no real crossing of the x-axis).

Worked Example

Solve x² − 5x + 6 = 0 (a = 1, b = −5, c = 6):

Check: 3² − 5(3) + 6 = 9 − 15 + 6 = 0 ✓ and 2² − 5(2) + 6 = 4 − 10 + 6 = 0 ✓.

The Vertex (the Turning Point)

The parabola's vertex sits at:

Plug that x back in to get the y-coordinate. It's a minimum when a > 0 and a maximum when a < 0 — handy for optimization problems.

Why It Matters in Real Life

Businesses use it to find the price that maximizes profit. Physicists use it to model the path of a ball. Anyone pricing a subscription, a flight or a loan is effectively working with a quadratic. Understanding the formula means understanding where that optimum sits — not just trusting a spreadsheet.

Skip the algebra when you're in a hurry with our free Quadratic Equation Solver, and refresh related skills with the Exponent Calculator.