Quadratic Formula, Slope, Distance & Half-Life Calculator

Compute real or complex quadratic roots, discriminant Δ, Cartesian slope & distance, and half-life decay.

Quadratic Roots (1x² + -5x + 6 = 0)
x₁ = 3, x₂ = 2
Discriminant Δ = 1 • Line Slope m = 2 (Dist = 8.9443)
Discriminant (b² − 4ac) 1
Slope & Distance ((1,2)→(5,10)) m = 2, d = 8.944
Half-Life Remaining (3 t₁/₂) 12.5%
Parabola Vertex x = −b/(2a) 2.5
Inputs
Coefficient of x² (must be non-zero).
Coefficient of x.
Constant y-intercept term.
Starting x coordinate.
Starting y coordinate.
Ending x coordinate.
Ending y coordinate.
Number of half-life periods elapsed.
Results & Graph
Quadratic Roots (1x² + -5x + 6 = 0)
x₁ = 3, x₂ = 2
Discriminant Δ = 1 • Line Slope m = 2 (Dist = 8.9443)
Discriminant (b² − 4ac)
1
Real roots
Slope & Distance ((1,2)→(5,10))
m = 2, d = 8.944
Half-Life Remaining (3 t₁/₂)
12.5%
Parabola Vertex x = −b/(2a)
2.5

Algebraic & Coordinate Magnitude Summary

Discriminant, Euclidean Distance, and Remaining Half-Life %
Graph 1 • Cartesian Parabola Plot y = ax² + bx + c Centered at axis of symmetry x = −b/(2a) = 2.50
-0.3 4.6 9.5 x=-0.5: 8.75 8.8 x=-0.5 x=0.5: 3.75 3.8 x=0.5 x=1.5: 0.75 0.8 x=1.5 x=2.5: -0.25 -0.3 x=2.5 x=3.5: 0.75 0.8 x=3.5 x=4.5: 3.75 3.8 x=4.5 x=5.5: 8.75 8.8 x=5.5
Graph 2 • Cartesian Vector (x₁, y₁) → (x₂, y₂), Slope & Midpoint Visualizes horizontal run Δx, vertical rise Δy, and hypotenuse d
P₁(1, 2) P₂(5, 10) Δx = 4 Δy = 8 Coordinate Geometry Analysis • Slope (m = Δy/Δx): 2.0000 • Euclidean Distance (d): 8.9443 • Midpoint M: (3.00, 6.00)
Graph 3 • Exponential Decay Remaining Percentage Across 0–6 Half-Lives 50% → 25% → 12.5% → 6.25% → 3.125% → 1.56%
0% 54% 108% 0 t½: 100 % 100 0 t½ 1 t½: 50 % 50 1 t½ 2 t½: 25 % 25 2 t½ 3 t½: 12.5 % 12.5 3 t½ 4 t½: 6.25 % 6.3 4 t½ 5 t½: 3.13 % 3.1 5 t½ 6 t½: 1.56 % 1.6 6 t½
Summary: Quadratic roots are x₁ = 3, x₂ = 2 and Euclidean distance is 8.9443.
Summary: For 1x² + (-5)x + (6) = 0, the discriminant is Δ = 1, yielding roots x₁ = 3, x₂ = 2. Between points (1, 2) and (5, 10), the Cartesian slope is 2 and Euclidean distance is 8.9443. After 3 half-life periods, 12.5% of a decaying quantity remains.

Formula & Mathematical Methodology: Quadratic Formula, Euclidean Metric & Exponential Decay

x = −b ± b2 − 4ac2a,   m = y2 − y1x2 − x1,   d = (x2 − x1)2 + (y2 − y1)2
x = [ −b ± √(b² − 4ac) ] / 2a; m = (y₂ − y₁)/(x₂ − x₁); d = √[(x₂−x₁)² + (y₂−y₁)²]; N(t) = N₀ × (1/2)^(t / t₁/₂)
  • Δ = b² − 4ac: Discriminant (>0 two real roots, =0 one repeated root, <0 complex conjugate roots)

Rounding: Rounded to 4 decimal places.

Worked Example: x² − 5x + 6 = 0 and Line from (1, 2) to (5, 10)

Find roots of x² − 5x + 6 = 0 and slope/distance between (1,2) and (5,10).

  1. Discriminant Δ = (−5)² − 4(1)(6) = 25 − 24 = 1 → Roots x₁ = 3, x₂ = 2.
  2. Slope m = (10 − 2) ÷ (5 − 1) = 8 ÷ 4 = 2; Distance d = √(4² + 8²) = √80 = 8.9443.
  3. After 3 half-lives, (0.5)³ = 12.5% remains.
Result: Roots are x₁ = 3, x₂ = 2; slope m = 2, distance d = 8.9443.

Frequently Asked Questions

What does a negative discriminant mean in the quadratic formula?

When b² − 4ac < 0, the parabola does not cross the real x-axis; its roots are a pair of complex conjugates a ± bi.

Mathematics & Statistics • Algebra & Coordinate Geometry Guide

How to Use the Quadratic Formula, Slope, Distance & Half-Life Calculator: Practical Guide & Visual Analysis

Solve second-degree polynomials ax² + bx + c = 0 with full discriminant analysis, measure gradient and distance between 2D coordinates, and calculate exponential decay over half-life intervals. Whether you are researching quadratic formula calculator, slope calculator, distance calculator, half life calculator, number sequence calculator, this interactive tool helps you solve quadratic equations (real and complex roots), compute Cartesian slope and Euclidean distance between two points, and evaluate exponential half-life decay.

Interpreting the 3 Custom Visual Graphs & Key Outputs

As you adjust Quadratic Coefficient (a), Linear Coefficient (b), Constant Term (c), Point 1 X-Coordinate (x₁), the calculator immediately computes Quadratic Roots x₁, x₂ (Real or Complex), Discriminant (b² − 4ac) & Parabola Vertex, Cartesian Slope (m) & Euclidean Distance (d), Remaining Half-Life Fraction (%) and updates all three domain-specific SVG charts in real time. Instead of relying on a single static figure, you can inspect the primary domain diagram, component breakdown, and multi-point trajectory curve to see how changing your inputs shifts the final result. Use the Compare (up to 3 scenarios) button to evaluate side-by-side variations before making a decision.

Real-World Benchmark & Planning Tips (Global)

In our benchmark scenario (x² − 5x + 6 = 0 and Line from (1, 2) to (5, 10)), Roots are x₁ = 3, x₂ = 2; slope m = 2, distance d = 8.9443. Keep in mind that uses Cartesian planar coordinates. For related calculations in Mathematics & Statistics, explore Scientific Calculator, Geometry Calculator (Right Triangle, Circle & Cylinder), Fraction, Mixed Number, GCF, LCM & Prime Factorization Calculator.