The Formula
Every equation of the form ax² + bx + c = 0 with a ≠ 0 is solved by:
x = (−b ± √(b² − 4ac)) / 2a
It comes from completing the square on the general equation, and it always works — unlike factoring, which only helps when the roots happen to be rational.
Discriminant Chart
The expression under the root, b² − 4ac, determines what kind of solutions exist before you compute them:
| Discriminant | Roots | Graph |
|---|---|---|
| Positive | Two distinct real roots | Crosses the x-axis twice |
| Zero | One repeated real root | Touches the x-axis at the vertex |
| Negative | Two complex conjugate roots | Never meets the x-axis |
A positive discriminant that is also a perfect square means the roots are rational, and the quadratic factors neatly over the integers.
The Vertex and the Parabola
The graph is a parabola opening upward when a > 0 and downward when a < 0. Its vertex sits at x = −b/2a, which is also the axis of symmetry — exactly midway between the two roots when they exist.
The vertex is the minimum or maximum of the function, which is why quadratics appear in every optimisation problem with a single squared term: projectile height, profit against price, area for a fixed perimeter.
Vieta's Relations
Without solving anything, the roots satisfy:
- Sum of the roots = −b/a
- Product of the roots = c/a
These give a fast check on any answer, and they let you construct a quadratic with specified roots.
Where Quadratics Come From
Anything involving a squared quantity: projectile motion under gravity, area problems, the relationship between speed and braking distance, resonance in circuits, and optimal pricing where both price and quantity vary. The Babylonians were solving quadratic problems by around 2000 BC, though negative and complex roots took another three thousand years to be accepted.
Frequently Asked Questions
What if a = 0?
The equation is linear, not quadratic, and the formula divides by zero. Solve bx + c = 0 directly, giving x = −c/b.
What does an imaginary root mean physically?
Usually that the situation described cannot occur — a projectile that never reaches the specified height, for instance. In electrical engineering and quantum mechanics, complex roots have direct physical meaning as oscillation and phase.
Is completing the square still worth learning?
Yes. It derives the formula, converts to vertex form, and appears throughout calculus and analytic geometry. The formula is the shortcut; completing the square is the method behind it.