Algebra & Mathematical Systems Calculator
Solve quadratic & polynomial roots, evaluate multi-variable linear systems via Cramer’s rule, and analyze exponential-logarithmic models
Polynomial Coefficients: a·x² + b·x + c = 0
Equation Roots / Solutions
x₁ = 3.000, x₂ = 2.000
Two Distinct Real Roots
Factored: (x – 3.000)(x – 2.000) = 0
Discriminant (Δ = b² – 4ac)
1.000
Δ > 0 (Two distinct real intersections)
Parabola Vertex (h, k)
(2.500, -0.250)
Axis of symmetry: x = 2.500
Vertex Form Representation
y = (x – 2.500)² – 0.250
y = a(x – h)² + k
Vieta’s Properties (Sum & Product)
Sum = 5.000 • Prod = 6.000
Sum = -b/a • Product = c/a
Equation 1: a₁·x + b₁·y = c₁
Equation 2: a₂·x + b₂·y = c₂
System Vector Solution
x = 1.000, y = 2.000
Consistent & Independent
Matrix Determinant (det A): -17.000
Primary System Determinant (D)
-17.000
Non-zero ensures invertible matrix
Variable Determinant (Dx)
-17.000
x = Dx / D = 1.000
Variable Determinant (Dy)
-34.000
y = Dy / D = 2.000
Variable Determinant (Dz)
N/A (2×2 System)
Residual: |Ax – b| ≈ 0.000
Solved Exponent / Variable (x)
x = 4.000
Equation: 5.000 · 2^(1.000 · x) = 80.000
Natural Log Equivalent (ln)
ln(C/A) = 2.773
Base-e Napierian logarithm
Common Log Equivalent (log₁₀)
log₁₀(C/A) = 1.204
Standard base-10 decadic scale
Half-Life / Doubling Period
T = 0.693 / k
Time required for 2× or 0.5× factor
Continuous Equivalent Rate (r)
+69.31%
Effective instantaneous growth rate