02 // Visual Math & Finance SolversΔ = b² − 4ac, x₁ + x₂ = −b/a, x₁·x₂ = c/a
Visual Quadratic Discriminant ($\Delta = b^2 - 4ac$) & Root Nature Analyzer
The quadratic discriminant $\Delta = b^2 - 4ac$ determines root multiplicity: $\Delta > 0$ yields two distinct real roots, $\Delta = 0$ yields one repeated real double root at the vertex, and $\Delta < 0$ yields two complex conjugate roots $-\frac{b}{2a} \pm i\frac{\sqrt{|\Delta|}}{2a}$.
Δ = b² − 4ac, x₁ + x₂ = −b/a, x₁·x₂ = c/aInputs & Parameters (3)
-4±0.254
-10±0.510
-12±0.512
Discriminant Δ = b² − 4ac
0
1 Repeated Real Root
Roots (Zero Crossings)
x = 2 (Double Root)
Sum = 4 | Product = 4
Vertex (h, k)
(2, 0)
Global Minimum (Opens Up)
Axis & Y-Intercept
x = 2
Y-Intercept = (0, 4)
20-Row Comparison Table
Compare across:
| # | Curvature (a) | Discriminant (Δ = b² − 4ac) | Vertex (h, k) | Root Nature |
|---|---|---|---|---|
| 01 | -4 | 80 | (-0.5, 5) | 2 Real |
| 02 | -3.5 | 72 | (-0.57, 5.14) | 2 Real |
| 03 | -3.25 | 68 | (-0.62, 5.23) | 2 Real |
| 04 | -2.75 | 60 | (-0.73, 5.45) | 2 Real |
| 05 | -2.25 | 52 | (-0.89, 5.78) | 2 Real |
| 06 | -2 | 48 | (-1, 6) | 2 Real |
| 07 | -1.5 | 40 | (-1.33, 6.67) | 2 Real |
| 08 | -1 | 32 | (-2, 8) | 2 Real |
| 09 | -0.75 | 28 | (-2.67, 9.33) | 2 Real |
| 10 | -0.25 | 20 | (-8, 20) | 2 Real |
| 11 | 0.25 | 12 | (8, -12) | 2 Real |
| 12 | 0.75 | 4 | (2.67, -1.33) | 2 Real |
| 13 | 1 (Current) | 0 | (2, 0) | 1 Real |
| 14 | 1.5 | -8 | (1.33, 1.33) | 2 Complex |
| 15 | 2 | -16 | (1, 2) | 2 Complex |
| 16 | 2.25 | -20 | (0.89, 2.22) | 2 Complex |
| 17 | 2.75 | -28 | (0.73, 2.55) | 2 Complex |
| 18 | 3.25 | -36 | (0.62, 2.77) | 2 Complex |
| 19 | 3.5 | -40 | (0.57, 2.86) | 2 Complex |
| 20 | 4 | -48 | (0.5, 3) | 2 Complex |
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Key Analytical Takeaways
- 01At default $a=1, b=-4, c=4$, the discriminant is $\Delta = (-4)^2 - 4(1)(4) = 0$, placing the vertex tangentially on the x-axis at $x = 2$.
- 02Increasing $c$ above $4$ pushes $\Delta < 0$, lifting the parabola above the x-axis and displaying the exact complex conjugate pair $u \pm vi$.
- 03By Vieta's Formulas, the sum of the two roots is always $-b/a$ and their product is always $c/a$.
Frequently Asked Questions
Why does a negative discriminant produce complex conjugate roots?+
When b² − 4ac < 0, taking the square root √(b² − 4ac) involves the imaginary unit i = √(−1), producing a symmetric pair of complex roots u + vi and u − vi.
What are Vieta's Formulas for a quadratic?+
Without even solving for the roots x₁ and x₂, their sum is guaranteed to be x₁ + x₂ = −b/a and their product is x₁·x₂ = c/a.
What are Vieta's formulas for a quadratic equation?+
Without computing radicals, Vieta's formulas relate the two roots directly to coefficients: their sum is x₁ + x₂ = −b / a and their product is x₁ · x₂ = c / a.