2026-09-29 10 min

Polynomial Equation Calculator

Solve the degree-2 polynomial equation ax² + bx + c = 0: factored hint, discriminant, roots and vertex. Free polynomial solver.

Calculator

Polynomial Coefficients

Lead coeff a

Coeff b

Constant c

Results update live as you type

Status:2x² + -7x + 3 = 0x₁ = 3 · x₂ = 0.5Two distinct real zeros

Polynomial Summary

Zerosx₁ = 3 · x₂ = 0.5

Factored2(x − (3))(x − (0.5))

Vertex(1.75, -3.125)

DiscriminantΔ = 25

Results

Optimal

Zero x₁

First root

Zero x₂

0.5

Second root

Zero Nature

Two distinct real zeros

Real vs complex classification

Factored Form

2(x − (3))(x − (0.5))

a(x − r₁)(x − r₂)

Vertex

(1.75, -3.125)

Extreme point of the graph

Discriminant

b² − 4ac

Roots verified by substitution and Vieta relations; factored form expands back to the original within rounding.

Visualizations

Coefficient Magnitudes

Root Reality

Two distinct real zeros100%

Formula

A degree-2 polynomial splits into two linear factors over the complex numbers; the discriminant tells whether the factors are real.

ax² + bx + c = a(x − r₁)(x − r₂) with r from the quadratic formula

Alternative Form

Vertex read: h = −b/2a, k = c − b²/4a

Variables

Symbol	Name	Unit	Description
a	Leading coefficient	number	Scales and opens the parabola
b	Middle coefficient	number	Shifts the axis
c	Constant	number	y-intercept
r₁, r₂	Zeros	numbers	Points where the polynomial vanishes

Explanation

Zeros are x-intercepts; factored form wears them openly; the vertex sits midway between real zeros.

Derivation

Factor theorem: r is a zero ⟺ (x − r) divides the polynomial. Quadratics yield r₁, r₂ via (−b ± √Δ)/2a.

Step-by-Step Calculation

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

Step 1

Write the polynomial equation in standard form.

ax² + bx + c = 0

eq:2x² + -7x + 3 = 0

Result: Standard form stated

Step 2

Compute the discriminant.

Δ = b² − 4ac

disc:25

Intermediate: -7² − 4(2)(3) = 25

Result: Two distinct real zeros

Step 3

Extract the zeros from the discriminant sign.

x = (−b ± √Δ)/2a

x1:3

x2:0.5

Result: x₁ = 3, x₂ = 0.5

Step 4

Write the factored form.

a(x − r₁)(x − r₂)

factored:2(x − (3))(x − (0.5))

Result: 2(x − (3))(x − (0.5))

Step 5

Locate the vertex for the graph story.

h = −b/2a, k = c − b²/4a

vertex:(1.75, -3.125)

Result: Vertex (1.75, -3.125)

Final Answer

Worked Examples

See how the calculator works with different scenarios.

Scenario

Non-monic quadratic.

Inputs

a:2

b:-7

c:3

Calculation

Δ = 49 − 24 = 25 → x = (7 ± 5)/4

Result

x₁ = 3, x₂ = 0.5; 2(x−3)(x−0.5)

Interpretation

Factored form expands back exactly.

Understanding Your Result

Your Result: Degree-2 polynomial solved with zeros, factors and vertex

Typical range: Δ > 0 two real zeros; Δ = 0 double; Δ < 0 complex pair; a = 0 linear.

What This Means

Zeros are where the curve meets the x-axis; factors encode them; the vertex is the extreme value point.

Why It Matters

Area, motion and profit models are quadratics — zeros give break-evens, vertices give optima.

Professional Interpretation

Read factored form as the answer and standard form as the question: converting between them is the skill.

Recommendations

•Compute Δ before factoring attempts
•Write factors with (x − r) signs
•Expand factors to double-check
•Note vertex for max/min context
•Handle a = 0 as linear

Educational Content

Introduction

Polynomial equations set a polynomial to zero; degree 2 brings the discriminant, factoring and the vertex together.

Concept Explanation

The factor theorem links zeros to factors; completing the square links standard form to the vertex and the formula.

Applications

▸Break-even analysis

▸Projectile landing times

▸Area with fixed perimeter

▸Profit maximization

▸Beam deflection zeros

▸Antenna tuning curves

Benefits

•Factored form produced
•Vertex included
•Vieta-style audit implicit
•Linear fallback
•Graph-ready outputs

Limitations

•Degree 2 focus
•Single variable
•Numeric coefficients

Best Practices

•Standardize first
•Try Δ before guessing factors
•Write (x − r) carefully
•Verify by expansion
•Sketch vertex + zeros

Common Misconceptions

Myth: Factors use (x + r).

Truth: They use (x − r): zero at +3 means factor (x − 3).

Myth: Δ < 0 means no zeros.

Truth: No real zeros — a complex pair still exists.

Myth: Vertex is always midway.

Truth: Midway between real zeros, yes — and always at −b/2a.

Myth: Leading a only stretches.

Truth: Its sign flips opening direction too.

Practical Tips

•Δ first, factors second
•Mind (x − r) signs
•Expand to verify factors
•Use vertex for extremes
•Spot perfect squares via Δ = 0
•Adapt when a = 0
•Keep standard form tidy

Common Mistakes

Writing (x + r) for positive roots

Zero r needs factor (x − r).

Guessing factors with big coefficients

Use the formula when guessing stalls.

Ignoring the leading a in factors

a(x − r₁)(x − r₂) keeps the a.

Vertex/root confusion

Vertex is the peak; zeros are crossings.

Forcing quadratics on linear input

a = 0 solves linearly.

Assumptions

This calculator makes the following assumptions to provide estimates:

Degree ≤ 2: Equation ax² + bx + c = 0.

Real coefficients: a, b, c real.

Exact factor link: Factor theorem applies.

Principal root: √ ≥ 0 for Δ ≥ 0.

Rounded display: Six decimals.

Limitations

Be aware of these limitations when using this calculator:

Degree 2: Cubics+ have own pages.

One variable: No systems.

Numeric: No parameters.

Illustrative graph: SVG window, not plotter.

Frequently Asked Questions

Answers to common questions about this calculator.

How do I solve 2x² − 7x + 3 = 0?

Δ = 25, so x = (7 ± 5)/4: x₁ = 3, x₂ = 0.5. Factored: 2(x − 3)(x − 0.5).

What is the factored form?

Rewriting as a(x − r₁)(x − r₂) so the zeros read off directly.

How are zeros and factors linked?

By the factor theorem: r zeroes the polynomial exactly when (x − r) divides it.

What does the vertex tell me?

The extreme point — maximum profit, minimum cost, or peak height for the same coefficients.

When is factoring faster than the formula?

With small integer roots visible by inspection; otherwise the formula wins.

What if a = 0 here?

The equation is linear bx + c = 0, solved directly as x = −c/b.

How do I verify factored form?

Expand it: the product must reproduce the original standard form.

Where do polynomial zeros matter?

Break-evens, landing times, resonance points — anywhere a model crosses zero.

Related Calculators

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Related Guides

Deepen your understanding with these educational resources.

Factoring fluency

From guess to guarantee.

Zeros on graphs

Intercepts with meaning.

Vertex tactics

Optima without calculus.

Degree awareness

Match method to degree.

Glossary

Technical terms used in mortgage calculations explained in plain language.

Polynomial: Sum of powers with numeric coefficients.
Zero (root): x making the polynomial zero.
Factor theorem: Zeros ↔ linear factors link.
Factored form: a(x − r₁)(x − r₂).
Standard form: ax² + bx + c = 0.
Vertex: Extreme point of the graph.
Discriminant: Δ classifying the zeros.
Double zero: Repeated root at Δ = 0.
Degree: Highest power present.

Authoritative References

Trusted sources and standards referenced by this calculator.

Khan Academy — Polynomials MathWorld — Polynomial Purplemath — Factoring quadratics Wikipedia — Quadratic equation Math is Fun — Polynomials Cuemath — Polynomial equations

Applicable Standards

Common Core HSA-APR.B — zeros and factorsNCTM Algebra — polynomial structure

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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.