Solve any quadratic equation with real or complex roots, vertex and discriminant.
The quadratic formula solves ax² + bx + c = 0 for any real coefficients. The discriminant b² − 4ac determines the nature of the roots: positive gives two distinct real roots, zero gives one repeated root, negative gives two complex conjugate roots. The vertex is the highest or lowest point of the parabola y = ax² + bx + c.
Quadratic formula
x = (−b ± √(b²−4ac)) / (2a)
Vertex
Vertex = (−b/(2a), −D/(4a))
The discriminant D = b² − 4ac reveals the nature of the roots without solving: D > 0 means two real roots; D = 0 means exactly one real root (the parabola touches the x-axis at its vertex); D < 0 means no real roots — the parabola doesn't cross the x-axis and the roots are complex.
When D < 0, the roots involve √(negative number) = imaginary numbers. They appear in conjugate pairs: a + bi and a − bi. They are mathematically valid but correspond to no real x-axis crossing.
Substitute each root back into ax² + bx + c. The result should be 0 (or very close to 0 due to floating-point precision). You can also verify by multiplying the factored form: a(x − x₁)(x − x₂) should expand to ax² + bx + c.