The quadratic formula
Every quadratic equation can be written as ax² + bx + c = 0 with a ≠ 0, and its solutions are:
x = (−b ± √(b² − 4ac)) / 2a
The expression under the square root, Δ = b² − 4ac, is called the discriminant, and it tells you how many solutions there are before you calculate them:
- Δ > 0: two distinct real roots (the parabola crosses the x-axis twice).
- Δ = 0: one repeated root (the parabola touches the x-axis at its vertex).
- Δ < 0: no real roots; there are two complex conjugate roots.
Example · 2x² − 5x − 3 = 0
Δ = (−5)² − 4·2·(−3) = 25 + 24 = 49. So x = (5 ± 7) / 4, which gives x₁ = 3 and x₂ = −1/2. Factored: 2(x − 3)(x + 1/2).
Incomplete quadratic equations
If a term is missing you do not need the quadratic formula:
- No constant term (c = 0): ax² + bx = 0 → x(ax + b) = 0 → x = 0 or x = −b/a.
- No x term (b = 0): ax² + c = 0 → x² = −c/a → x = ±√(−c/a), provided −c/a ≥ 0.
The vertex of the parabola
The graph of y = ax² + bx + c is a parabola with its vertex at x = −b/2a. If a > 0 it opens upward and the vertex is a minimum; if a < 0 it opens downward and the vertex is a maximum. The y-coordinate of the vertex is −Δ/4a.
Frequently asked questions
How do I check my answers?
Substitute them into the equation: they should give 0. You can also use Vieta's formulas: x₁ + x₂ = −b/a and x₁·x₂ = c/a.
What is the i in complex solutions?
The imaginary unit, defined by i² = −1. It lets you take square roots of negative numbers: √−16 = 4i.
Can I solve by factoring or completing the square instead?
Yes, all three methods give the same roots. Factoring is fastest when the roots are whole numbers; completing the square is how the quadratic formula is derived. The calculator shows the factored form when it exists.
What if I have a system of equations?
If they are linear, use the system of equations solver with Gaussian elimination.