@top_maths_5m: The sign of a quadratic, in one single journey ✨ Everything starts from ax² + bx + c with a > 0: the parabola opens upward. Its sign depends on one number only, the discriminant Δ = b² − 4ac. A comet leaves Δ and visits all three cases. 🔹 Δ > 0: two distinct roots, r₁ = (−b − √Δ)/(2a) and r₂ = (−b + √Δ)/(2a). The parabola crosses the x-axis at two points. The quadratic is positive outside the roots and negative between them. → ax² + bx + c > 0 ⟺ x ∈ (−∞, r₁) ∪ (r₂, +∞) 🔹 Δ = 0: one double root, r = −b/(2a). The parabola touches the x-axis at a single point. The quadratic is positive everywhere, except at r, where it equals zero. → x ∈ (−∞, r) ∪ (r, +∞) 🔹 Δ < 0: no real root. The parabola stays above the x-axis. → x ∈ ℝ 🔄 What if a < 0? The parabola flips: same roots, opposite signs. For Δ > 0, the quadratic is positive only between the roots. For Δ ≤ 0, the inequality has no solution. 💡 Key idea: every sign in the table is the vertical projection of a piece of the curve. Reading the parabola means reading the sign table. An animation built step by step, with rigor and beauty as the only rules. #maths #math #quadratic #parabola #animation

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2026-11-08 02:22:05

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