2026-09-29 9 min

Quadratic Programming Calculator

Free quadratic programming calculator: minimize ax² + bx subject to l ≤ x ≤ u with unconstrained minimizer, projection, active bound and value.

Calculator

Objective and Box

Coefficient a

Coefficient b

Lower bound l

Upper bound u

Results update live as you type

Status:min 2x² + -12x on [0, 5]x* = 3, f = -18

Solution Summary

Free minimizerx_free = 3

Clipped optimizerx* = 3 (interior (no bound binds))

Minimum-18

Results

Optimal

Constrained minimum

-18

At x* = 3

Optimizer x*

Clipped free minimizer

Free minimizer

-b / 2a ignoring bounds

Active set

interior (no bound binds)

Convex: clip is optimal

Results are recomputed live and verified against the step-by-step work below.

Visualizations

Candidates

0f(l)

-18f(x*)

-10f(u)

Formula

Convex quadratics pull toward one point; bounds simply clip that pull to the box.

min ax² + bx s.t. l ≤ x ≤ u; x_free = -b / 2a, x* = clamp(x_free, l, u)

Alternative Form

Interior solution when x_free sits inside; otherwise the nearer bound binds

Variables

Symbol	Name	Unit	Description
a, b	Objective coeffs	numbers	a > 0 gives convex pull
l, u	Box bounds	numbers	Lower and upper walls
x_free	Free minimizer	number	-b / 2a ignoring bounds
active	Active set	text	Interior, lower, or upper

Explanation

Compute the free minimizer, clip it into the box, and evaluate — the clip tells which bound (if any) binds.

Derivation

Derivative 2ax + b = 0 gives x_free; convexity (a > 0) makes clipping optimal by monotonicity outside x_free.

Step-by-Step Calculation

Follow each step to understand how the result is calculated from your inputs.

Step 1

Write the QP.

min ax² + bx, l ≤ x ≤ u

qp:2x² + -12x on [0, 5]

Result: Model stated

Step 2

Solve the free problem: 2ax + b = 0.

x_free = -b / 2a

xf:3

Result: x_free = 3

Step 3

Clip into the box.

x* = clamp

xs:3

Intermediate: min(5, max(0, 3))

Result: x* = 3 (interior (no bound binds))

Step 4

Evaluate the objective at x*.

f(x*)

fv:-18

Result: f = -18

Step 5

Compare against both walls to confirm.

Wall check

fl:0

fu:-10

Result: Best confirmed

Final Answer

-18

Worked Examples

See how the calculator works with different scenarios.

Scenario

Free point inside.

Inputs

a:2

b:-12

l:0

u:5

Calculation

x_free = 3 inside, f = -18

Result

x* = 3, f = -18

Interpretation

No bound binds.

Understanding Your Result

What This Means

The headline result is the exact answer for the quadratic programming calculator inputs you entered, with supporting values shown beside it.

Why It Matters

Homework, exams and real-world models all reduce to this calculation; a verified numeric answer beats guessing.

Professional Interpretation

Read the headline result first, then scan the supporting rows and the step list to confirm the arithmetic before reusing the value.

Recommendations

•Verify the headline result against the steps
•Test one worked example by hand
•Check edge cases separately
•Keep full precision until the final display
•Restate the answer in the problem context

Educational Content

Introduction

This quadratic programming calculator evaluates your inputs with closed-form arithmetic and shows every intermediate value.

Concept Explanation

Compute the free minimizer, clip it into the box, and evaluate — the clip tells which bound (if any) binds.

Applications

▸Homework and exams

▸Engineering checks

▸Finance models

▸Science labs

▸Programming logic

▸Everyday estimation

Benefits

•Instant verified answer
•Step-by-step work shown
•Charts that expose mistakes
•Handles decimals directly
•Free with examples

Limitations

•Numeric inputs only
•Real-number results
•Display rounding to 6 decimals

Best Practices

•Enter numbers carefully
•Check the steps line by line
•Verify with one example
•Watch units in word problems
•State the final answer with context

Common Misconceptions

Myth: One input is enough.

Truth: Most problems need every coefficient; missing values change the answer.

Myth: Rounding early is fine.

Truth: Keep full precision until the final display to avoid drift.

Myth: The chart is decoration.

Truth: The chart is a second computation — mismatched bars mean a bad input.

Myth: Any rearrangement works.

Truth: Only equivalent operations preserve equality; follow the steps.

Practical Tips

•Double-check each input
•Read the steps in order
•Compare against one worked example
•Verify edge cases separately
•Keep units consistent
•Use Reset to restore defaults
•Copy the summary for notes

Common Mistakes

Typing a value in the wrong field

Match each label to the formula variables before calculating.

Dividing by zero

Check denominators and leading coefficients first.

Ignoring the steps

The step list shows exactly where a sign or arithmetic slip hides.

Rounding mid-calculation

Round only the displayed answer.

Forgetting to verify

Substitute the answer back into the original statement.

Assumptions

This calculator makes the following assumptions to provide estimates:

Real numbers: Inputs and outputs are real numbers.

Exact inputs: Coefficients are taken as given.

Standard notation: Conventional symbols and operator precedence.

Display rounding: Six decimals in outputs.

Limitations

Be aware of these limitations when using this calculator:

Numeric scope: Symbolic parameters need the advanced pages.

Single scenario: One input set per calculation.

No units conversion: Convert units before entering values.

Frequently Asked Questions

Answers to common questions about this calculator.

What is quadratic programming?

Minimizing a quadratic subject to linear bounds — here the 1D box version with an exact clip solution.

Why does clipping work?

A convex quadratic falls toward x_free and rises after, so the feasible point nearest x_free wins.

What does active set mean?

Which bounds bind: interior means none, otherwise the nearer wall decides.

What if a < 0?

The objective is concave (a peak, not a bowl); the minimum sits at a wall, and the page suggests checking endpoints.

How is this different from plain optimization?

Plain optimization ignores bounds; QP enforces l ≤ x ≤ u.

How do I verify?

Evaluate f at l, u, and x_free (if inside) and confirm x* is lowest.

Related Calculators

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Related Guides

Deepen your understanding with these educational resources.

Showing your steps

Write one operation per line.

Checking answers

Substitute back every time.

Reading interval notation

Brackets include, parentheses exclude.

Word-problem setup

Name variables before computing.

Glossary

Technical terms used in mortgage calculations explained in plain language.

Solution set: All values satisfying the given statement.
Interval notation: Compact bracket notation for ranges of the number line.
Critical value: A point where the expression is zero or undefined, splitting sign regions.
Verification: Substituting the answer back to confirm the original statement.
Closed form: An exact formula evaluated directly, not an approximation loop.
Edge case: Inputs where the usual rule breaks, such as zero denominators.

Authoritative References

Trusted sources and standards referenced by this calculator.

Khan Academy — Math foundations Math is Fun — Index Purplemath — Lessons Wikipedia — Mathematics Cuemath — Math resources

Applicable Standards

Common Core HSA — creating and reasoning with equationsNCTM Algebra — representations and equivalence

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Page Information

Last Updated:

2026-09-29

Version:

2.0.0

References:

License:

Free to use for educational and professional purposes.