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.
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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

Quadratic Formula, Slope, Distance & Half-Life Calculator: inputs, method, and how to read the result

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

Choose inputs for the case you are checking

The defaults are illustrative. Replace them with values for your case and keep the units shown beside each field.

  • Quadratic Coefficient (a): Coefficient of x² (must be non-zero).
  • Linear Coefficient (b): Coefficient of x.
  • Constant Term (c): Constant y-intercept term.
  • Point 1 X-Coordinate (x₁): Starting x coordinate.

What the calculation includes

With the default values shown, Quadratic Roots (1x² + -5x + 6 = 0) is x₁ = 3, x₂ = 2. The calculation follows Quadratic Formula, Euclidean Metric & Exponential Decay; the formula and variable definitions above show how the inputs combine.

One-input check: With the other defaults held constant, changing Quadratic Coefficient (a) from 1 to 2 changes Quadratic Roots (1x² + -5x + 6 = 0) (x₁ = 3, x₂ = 2) to Quadratic Roots (2x² + -5x + 6 = 0) (1.25 ± 1.199i). This is a comparison inside the calculator's stated model, not a prediction.

Where to use caution

  • Uses Cartesian planar coordinates.
  • Vertical lines (x₁ = x₂) have undefined (infinite) slope.

Scope: Global Check the assumptions and limitations against your situation before relying on the estimate.

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