Math & Statistics
Quadratic Newton Approximation Calculator
Generate Newton estimates, residuals, derivatives, and reference-root errors from an entered initial guess.
f(x) at initial guess-
f'(x) at initial guess-
First Newton estimate-
First estimate minus reference root-
Polynomial value after one step-
Derivative at first estimate-
Second Newton estimate-
Second estimate minus reference root-
Decision view
Newton tangent-intersection diagram
Newton tangent-intersection diagramThe quadratic curve, tangent lines, and successive x-intercepts expose the iteration geometry.
Exact scenario comparisonInitial x guess changes while all other entered assumptions remain constant.
| Initial x guess | f(x) at initial guess | f'(x) at initial guess | First Newton estimate | First estimate minus reference root | Polynomial value after one step | Derivative at first estimate | Second Newton estimate | Second estimate minus reference root |
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How to use Quadratic Newton Approximation Calculator
- Enter the quadratic and initial guess.
- Use the reference root only as an error comparison.
- Inspect the first and second tangent intersections and their residuals.
Calculator guide
Understanding Quadratic Newton Approximation Calculator
Newton's method repeatedly intersects a tangent line with the x-axis, so convergence should be shown as a geometric process rather than a generic flow.
Calculation method
How the calculation works
Perform two transparent Newton updates from the entered initial guess and compare both estimates with a separately entered reference root. Evaluate f(x) and f'(x), then perform two explicit updates x_next=x-f(x)/f'(x).
Detailed calculation process
Apply Newton updates to the entered quadratic
The default solves x²-2x-3 from x₀=4, compares with reference root 3, and displays two explicit Newton updates.
General formula: f(x) = ax^2+bx+cf'(x) = 2ax+bx_(k+1) = x_k-f(x_k)/f'(x_k)e_k = x_k-r_refrho_k = f(x_k) Each tangent at the current estimate crosses the x-axis at the next estimate. A small derivative can make the update unstable or undefined.
What each symbol means
Worked substitution with the default inputs
The default moves 4 → 3.166667 → 3.006410 in two steps, reducing the reference-root error to about 0.00641.
Purpose-built visual
Newton tangent-intersection diagram
The quadratic curve, tangent segments, vertical projections, and successive x-intercepts show the actual iteration geometry.
Worked situations
Practical examples
- The default solves x²-2x-3 from x₀=4, compares with reference root 3, and displays two explicit Newton updates.
- The default moves 4 → 3.166667 → 3.006410 in two steps, reducing the reference-root error to about 0.00641.
Better inputs
Useful tips
- Change one input at a time and confirm both the result and visual move.
- Keep the units stated beside every field.
- Retain intermediate precision and round only the reported result.
Before relying on the result
Limitations and common mistakes
- Newton's method can fail at a zero or very small derivative.
- It can converge to another root or diverge from a poor guess.
- The reference root does not drive the update.
Reference
Key terms
- Newton step
- Current estimate minus function value divided by derivative.
- Tangent intercept
- Where the current tangent crosses the x-axis.
- Residual
- Polynomial value at an estimate.
Important note
Calculated directly from the entered values using the displayed formula and rounding settings.
Frequently asked questions
Does the reference root affect iteration?
No.
Why can Newton fail?
The derivative can be too small or the guess can be poor.
Are two iterations always sufficient?
No; this page exposes two auditable steps, and further iteration may be required for a chosen tolerance.
Which root is found?
It depends on the initial guess and function.