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Quadratic Equation Calculator

v1.3

Solve ax² + bx + c = 0 with roots, discriminant, vertex and axis of symmetry.

Last updated · 1 min read

  • Formula verified
  • Editorially reviewed
  • Updated June 27, 2026
  • Privacy first

Version 1.3 · Uses official U.S. guidance where applicable · Free forever · No signup required · Works on mobile

Quick answer

What this calculator does
Quadratic Equation Calculator solve ax² + bx + c = 0 with roots, discriminant, vertex and axis of symmetry. It runs entirely in your browser, needs no signup, and uses standard published U.S. formulas.
When to use it
Use the Quadratic Equation Calculator whenever you need a fast, reliable quadratic equation estimate. It is an estimate, not a quote or professional advice.
Key inputs
  • Coefficient a (x² term)
  • Coefficient b (x term)
  • Constant c
Key outputs
  • Roots
  • Discriminant (b² − 4ac)
  • Vertex
  • Axis of symmetry
  • Sum of roots (−b/a)
  • Product of roots (c/a)
Takeaway
In one line: enter your coefficient a (x² term) and coefficient b (x term) and the Quadratic Equation Calculator returns roots and discriminant (b² − 4ac) you can compare before deciding.

Inputs

Cannot be 0.

Result

Calculation complete

Roots

x₁ = 2, x₂ = 1

Two distinct real roots

Discriminant (b² − 4ac)1
Vertex(1.5, -0.25)
Axis of symmetryx = 1.5
Sum of roots (−b/a)3
Product of roots (c/a)2
OpensUpward (minimum at vertex)

Key takeaway

1x² + -3x + 2 = 0 → x₁ = 2, x₂ = 1 (discriminant 1)

Saved calculations remain on this device unless you choose to share them. No tracking, no account needed.

Share this result

Shares the calculator link. Your inputs stay on your device.

Privacy-safe: amounts you typed are never attached to a share link.

My dashboard

Saved on this device only — no account needed, no financial values leave your browser.

Visual breakdown

The same numbers as above, shown visually so the trade-offs are easy to see at a glance.

Result breakdown
Roots
x₁ = 2, x₂ = 1
Sum of roots (−b/a)
3
Product of roots (c/a)
2
Vertex
(1.5, -0.25)
Axis of symmetry
x = 1.5
Discriminant (b² − 4ac)
1

Smart financial assistant

What this means

Roots is x₁ = 2, x₂ = 1. This result is a directional estimate based on the values you entered.

Is this good or bad?

A result is useful when the inputs reflect your real situation — re-run it as your numbers change.

What's the risk?

Estimates use standard assumptions; your actual outcome depends on the terms you are offered.

What should you do next?

Save or print this result, then confirm the key assumptions with an official source.

Your action plan

  1. 1

    Double-check the inputs you estimated

    The output is only as good as the least certain input.

  2. 2

    Compare at least three alternatives

    One quote is a data point, not a market.

  3. 3

    Save this result

    It gives you a baseline to compare against later.

Smart timeline

Today

  • Save or print this result.
  • Note which inputs you guessed at.

This month

  • Replace estimated inputs with real numbers and re-run.
  • Compare alternatives before committing.

Next 12 months

  • Re-run this calculation as your situation changes.
  • Track the outcome against this baseline.

Long term

  • Revisit the decision annually.
  • Adjust as rates, income or goals change.

Your next steps on this site

Result Intelligence

Understand your result

  • Your roots is x₁ = 2, x₂ = 1. Use it as a directional estimate; always confirm with an official source before acting.

What should you do next?

  1. Debt-to-Income Calculator

    Check whether lenders will approve you.

  2. Credit Utilization Calculator

    Lower utilization can improve your APR.

  3. Affordability Calculator

    See the maximum loan you should carry.

  4. Compound Interest Calculator

    Compare paying debt vs. investing the difference.

Ways to improve your result

  • Compare at least 3 alternatives before deciding.
  • Document assumptions so you can re-run this calculation as inputs change.
  • Consult a licensed professional before signing.

Common U.S. scenarios

U.S. household

Median assumptions used above; your city and credit tier will shift the numbers.

Result summary

Quadratic Equation Calculator

Quadratic Equation Calculator
Roots: x₁ = 2, x₂ = 1
Discriminant (b² − 4ac): 1
Vertex: (1.5, -0.25)
Axis of symmetry: x = 1.5
Sum of roots (−b/a): 3
Product of roots (c/a): 2
1x² + -3x + 2 = 0 → x₁ = 2, x₂ = 1 (discriminant 1)

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What this result assumes

Confidence in a number depends on the assumptions behind it. Here are ours, in full.

Assumptions

  • Results are rounded for display; internal math uses full precision.

Limitations

This is an estimate for planning and education, not a quote or professional advice.

Estimates are for education and planning. They are not financial, tax or legal advice.

Formula source

Standard mathematical identities

Version
v1.3
Last reviewed
MethodologyReferences

Where to go next

Next logical calculatorScientific CalculatorFull expression scientific calculator with trig, logs, roots, powers and factorials.Continue

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Instant result summary

Quadratic Equation Calculator: Your result appears above. It's a fast estimate based on the inputs you entered — use it to compare options, then confirm with a professional before making a decision.

What does this result mean?

  • The number is only as accurate as your inputs — small errors compound.
  • Compare 2–3 scenarios rather than acting on a single result.
  • Talk to a licensed professional before making a large financial or medical decision.

How does the formula work?

For ax² + bx + c = 0 with a ≠ 0, the quadratic formula gives x = (−b ± √(b² − 4ac)) / 2a. The discriminant D = b² − 4ac decides the nature of the roots: D > 0 gives two real roots, D = 0 gives one repeated root and D < 0 gives a complex conjugate pair. The vertex sits at x = −b/2a.

How it works

Enter the three coefficients exactly as they appear in standard form. The calculator computes the discriminant first, then applies the quadratic formula, and also reports the vertex, axis of symmetry and Vieta's relations (sum = −b/a, product = c/a) so you can check your work by hand.

Step-by-step guide

  1. 1Enter each input value into the calculator above.
  2. 2Review the result and any auto-generated breakdowns.
  3. 3Adjust inputs to compare scenarios.
  4. 4Copy, share, or print the results.

Example calculation

Example: x² − 3x + 2 = 0 factors as (x − 1)(x − 2), so the roots are 1 and 2.

Who should use this calculator?

  • Anyone who needs a fast, transparent estimate before making a decision.
  • Professionals validating a client-facing number.
  • Students learning the underlying formula.

When should you use it?

  • Any time you need a quick sanity check.
  • When comparing several scenarios side-by-side.
  • Before making a decision that depends on this number.

What affects the result?

  • Every input value you enter
  • The formula and assumptions displayed on this page
  • Rounding — small differences in inputs compound in the result

Common mistakes to avoid

  • Entering values in the wrong units.
  • Assuming the result includes taxes or fees when it may not.
  • Using rounded inputs and expecting exact figures.

Professional tips

  • Double-check your inputs before acting on the result.
  • Run 2–3 scenarios and compare — don't rely on a single number.
  • Save or print the results so you can review them later.

Advantages of using this calculator

  • Instant results with no signup, no tracking and no ads inside the calculator.
  • Transparent formulas — you can see exactly how the number is produced.
  • Save, share, print or embed your results anywhere.
  • Reviewed and updated regularly by our editorial team.

Limitations and caveats

  • Estimates only — real-world results depend on many factors we cannot see.
  • Not professional advice — consult a qualified expert before acting.
  • Assumptions are shown; changing them will change the result.

Key terms explained

Input
A value you enter into the calculator. Small differences in inputs can compound into large differences in the result.
Assumption
A fixed value the calculator uses when it isn't asked from you. Assumptions are documented in the formula section.
Estimate
A directional figure produced from your inputs and standard formulas. Always sanity-check before acting.

Before you act — decision checklist

  • Double-check every input for units and typos.
  • Compare 2–3 scenarios rather than trusting one result.
  • Confirm the assumptions in the formula match your situation.
  • Consult a qualified professional before acting on any large decision.

Trust & Accuracy

Formula verified

Verified against standard mathematical and financial conventions.

Last reviewed

June 27, 2026

Accuracy tested

Edge cases tested, formula validation and manual verification completed.

Editorially reviewed

Reviewed by our editorial team against our published methodology and re-checked whenever rates or limits change.

Educational use

This tool is provided for education and planning only. It is not financial, tax, legal or medical advice.

Available for

United States, Canada, United Kingdom, Australia, India, European Union.

Currency support

USD · INR · CAD · AUD · GBP · EUR

Privacy

No information entered into this calculator is stored on our servers unless you explicitly choose to save or share your calculation.

Print, share & save

Use the Save, Share, Copy, PDF, CSV and Print actions under the result panel.

Accessibility

Keyboard navigable, screen-reader friendly labels, responsive on mobile and desktop.

Learn more: How we verify formulas · How we test accuracy · Methodology · Editorial policy

Recent Updates

We continuously review and improve our calculators to keep formulas, assumptions and references accurate.

View change log (4)Show
  1. v1.3

    Added Trust & Accuracy panel and machine-readable change log.

  2. v1.2

    Added Save, Share, PDF, CSV and Print actions to results.

  3. v1.1

    Added global currency selector (USD, INR, CAD, AUD, GBP, EUR).

  4. v1.0

    Initial calculator release with verified formulas and Official Sources.

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Frequently Asked Questions

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Reviewed by the CreditLoanCalculator editorial team·Last reviewed ·Editorial policy·How we verify formulas
Disclaimer: Results are estimates for informational purposes only and should not be considered financial, medical, legal, or professional advice. Always consult a qualified professional before making decisions.