Find roots of quadratic and cubic polynomials with discriminant analysis and step-by-step solutions.
For ax² + bx + c = 0: x = (-b ± √Δ) / 2a, where discriminant Δ = b² - 4ac. • Δ > 0: Two distinct real roots • Δ = 0: One repeated root x = -b/2a • Δ < 0: Two complex conjugate roots
Solve ax² + bx + c = 0 using the quadratic formula: x = (-b ± √Δ) / 2a, where discriminant Δ = b² - 4ac. Example: x² - 5x + 6 = 0 (a=1, b=-5, c=6). Δ = 25 - 24 = 1 > 0. x = (5 ± 1) / 2, so x = 3 or 2. Factored form: (x-3)(x-2) = 0. If Δ is positive: two distinct real roots; zero: repeated root; negative: complex conjugate pair.
The discriminant Δ = b² - 4ac determines root type before solving. Δ > 0: Parabola crosses x-axis at two points. Δ = 0: Parabola touches x-axis (repeated root). Δ < 0: Parabola doesn't intersect x-axis (complex roots). The discriminant also relates to Vieta's formulas: sum of roots = -b/a, product of roots = c/a.
For y = ax² + bx + c, the vertex is (-b/2a, f(-b/2a)) and axis of symmetry is x = -b/2a. Example: y = x² - 4x + 3. Axis: x = 2. y(2) = -1, so vertex is (2, -1). The vertex represents the minimum (a>0) or maximum (a<0) value. Standard form: y = a(x - h)² + k where (h, k) is the vertex.
Cardano's Formula solves ax³ + bx² + cx + d = 0, discovered by 16th century mathematician Cardano. First convert to depressed cubic t³ + pt + q = 0 (substitution: t = x + b/3a). Calculate discriminant Δ = -4p³ - 27q². Δ > 0: three distinct real roots. Δ = 0: repeated roots. Δ < 0: one real root and two complex roots. Cubic equations always have at least one real root (intermediate value theorem).
Real-coefficient polynomials always have complex roots in conjugate pairs. If -1 + 2i is a root of x² + 2x + 5 = 0, then -1 - 2i is also a root. This occurs because when all coefficients are real, if complex a + bi is a root, then conjugate a - bi must also be a root. In complex plane, conjugate pairs are symmetric about the real axis.