The Quadratic Formula — Solving ax² + bx + c = 0
The quadratic equation ax² + bx + c = 0 appears throughout mathematics, physics, engineering, and economics. Its general solution — the quadratic formula x = (−b ± √(b² − 4ac)) / 2a — guarantees that any quadratic equation can be solved once you know its three coefficients. The parabola formed by y = ax² + bx + c intersects, touches, or misses the x-axis depending on whether the discriminant is positive, zero, or negative.
Discriminant D = b² − 4ac and root classification:
1. D > 0 — Two distinct real roots. The parabola crosses the x-axis at two separate points.
2. D = 0 — One repeated real root (double root): x = −b / 2a. The parabola is tangent to the x-axis.
3. D < 0 — No real roots; two complex conjugate roots. The parabola does not intersect the x-axis.
Vieta's formulas — relationships between roots and coefficients:
For roots x₁ and x₂: x₁ + x₂ = −b/a and x₁ × x₂ = c/a. These let you verify your answers without substituting back into the equation.
The coefficient a must not be zero (that would make the equation linear, not quadratic). Decimal and negative coefficients are fully supported.
Frequently Asked Questions
A: With a = 0 the equation becomes linear (bx + c = 0), not quadratic. The solver will display a warning and will not attempt a quadratic solution.
A: Yes. For example a = 0.5, b = −1.5, c = 1 are all valid inputs.
A: When D < 0, the roots are shown in the form (real part) ± (imaginary part)i.