AtlasCalcsUtility Calculation Suite
2026-09-29 9 min

Polynomial Root Calculator

Solve quadratic equations ax² + bx + c = 0 with the quadratic formula, discriminant analysis and verification. Free online root calculator.

Calculator

Quadratic Coefficients

a (x²) Help

Zero → linear mode

b (x) Help

Linear coefficient

c (constant) Help

Constant term

Results update live as you type
Status:1x² + -3x + 2 = 0x₁ = 2Two distinct real roots

Root Summary

x₁2
x₂1
Discriminant1
Vertex(1.5, -0.25)

Results

Optimal

Root x₁

2

First root of ax² + bx + c = 0

Root x₂

1

Second root (or conjugate)

Discriminant D

1

b² − 4ac decides root type

Root Nature

Two distinct real roots

Distinct, repeated, linear or complex

Vertex

(1.5, -0.25)

Turning point (−b/2a, f(−b/2a))

Vieta Check

sum = 3, product = 2

Sum −b/a and product c/a

Roots come from the exact quadratic formula in double precision and are verified against Vieta's sum and product relations.

Visualizations

Coefficient Magnitudes

1a
3b
2c
1|D|

Solution Completeness

Solved100%

|b²| vs |4ac|

b²9

Formula

Completing the square on ax² + bx + c = 0 isolates x and produces the quadratic formula; the discriminant under the root classifies the solutions.

x = (−b ± √(b² − 4ac)) / 2a

Alternative Form

Linear fallback: x = −c/b when a = 0

Variables

SymbolNameUnitDescription
aQuadratic coefficientnumberMultiplier of x²; zero means linear
bLinear coefficientnumberMultiplier of x
cConstant termnumberValue at x = 0
DDiscriminantnumberb² − 4ac; sign sets root type

Explanation

The ± splits one equation into two symmetric roots around −b/2a; the discriminant measures how far apart they are.

Derivation

Divide by a, complete the square: (x + b/2a)² = (b² − 4ac)/4a², then take square roots to get the formula.

Step-by-Step Calculation

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

Step 1

Write the equation in standard form.

ax² + bx + c = 0
a:1
b:-3
c:2
Result: 1x² + -3x + 2 = 0
Step 2

Compute the discriminant.

D = b² − 4ac
D:1
Intermediate: D = -3² − 4(1)(2) = 1
Result: D = 1 → Two distinct real roots
Step 3

Apply the quadratic formula.

x = (−b ± √D) / 2a
root1:2
root2:1
Intermediate: x = (3 ± √1) / 2
Result: x₁ = 2, x₂ = 1
Step 4

Locate the vertex (axis midpoint).

x_v = −b / 2a
vertex:(1.5, -0.25)
Result: Vertex at (1.5, -0.25)
Step 5

Verify with Vieta’s formulas.

sum = −b/a, product = c/a
check:sum = 3, product = 2
Result: sum = 3, product = 2
Final Answer
2

Worked Examples

See how the calculator works with different scenarios.

Scenario

Two friendly integer roots.

Inputs

a:1
b:-3
c:2

Calculation

D = 9 − 8 = 1; x = (3 ± 1)/2

Result

x₁ = 2, x₂ = 1

Interpretation

Factors as (x − 1)(x − 2).

Understanding Your Result

Your Result: Both roots with discriminant classification and checks

Typical range: D > 0: two reals; D = 0: one double; D < 0: complex pair.

What This Means

The roots are the x-values where the parabola (or line) crosses zero — the solutions of the equation.

Why It Matters

Roots locate break-even points, projectile landing spots, resonance frequencies, and factors of the polynomial.

Professional Interpretation

Read D as separation: large positive D means widely spaced crossings; D near zero means a grazing touch at the vertex.

Recommendations

  • •Check D sign before expecting reals
  • •Verify with Vieta sum and product
  • •Use vertex as midpoint sanity check
  • •Handle a = 0 as linear
  • •Substitute roots back to confirm zero

Educational Content

Introduction

This calculator solves any quadratic ax² + bx + c = 0, explaining each step from discriminant to verified roots.

Concept Explanation

Completing the square concentrates all x-dependence into one squared term, whose two square roots give the ± branches.

Applications

▸Projectile landing time
▸Profit break-even analysis
▸Lens and mirror equations
▸Circuit resonance
▸Area optimization
▸Factoring quadratics

Benefits

  • •Real and complex roots handled
  • •Discriminant explained numerically
  • •Vertex and axis included
  • •Vieta cross-check built in
  • •Linear fallback automatic

Limitations

  • •Quadratics (and linear) only
  • •Single variable x
  • •Display rounds to 6 decimals

Best Practices

  • •Write standard form first
  • •Compute D before rooting
  • •Keep ± on both branches
  • •Verify by substitution
  • •Sketch to confirm root count

Common Misconceptions

Myth: Every quadratic has two real roots.
Truth: Only D ≥ 0 gives real roots; D < 0 gives a complex pair.
Myth: D = 0 means no solution.
Truth: D = 0 gives exactly one (double) root at the vertex.
Myth: The formula fails when a = 0.
Truth: The equation is then linear — solve x = −c/b instead.
Myth: Complex roots are errors.
Truth: They are legitimate solutions; the parabola simply misses the x-axis.

Practical Tips

  • •Compute D first
  • •Keep both ± branches
  • •Verify with sum and product
  • •Vertex is the root midpoint
  • •Expect complex when D < 0
  • •Treat a = 0 as linear
  • •Substitute back to confirm

Common Mistakes

Dropping the ± branch
Always compute both −b + √D and −b − √D.
Dividing only part by 2a
The whole numerator (−b ± √D) is divided by 2a.
Sign error on −b
Negate all of b: if b = −3 then −b = +3.
Forgetting the linear fallback
Check a = 0 first and solve bx + c = 0.
Misreading complex roots
Split into real part −b/2a and imaginary part √|D|/2a.

Assumptions

This calculator makes the following assumptions to provide estimates:

Standard form: The equation is ax² + bx + c = 0.
Real coefficients: a, b, c are real numbers.
Single variable: Only x is unknown.
Exact inputs: Coefficients are treated as exact.
Display rounding: Results show up to 6 decimal places.

Limitations

Be aware of these limitations when using this calculator:

Degree ≤ 2: Cubics and higher need other solvers.
No symbolic factoring: Output is numeric roots, not factor steps.
Complex display: Complex roots shown as a ± bi text.
Rounding: Irrational roots are decimal approximations.

Frequently Asked Questions

Answers to common questions about this calculator.

How do I find the roots of a quadratic?
Use the quadratic formula x = (−b ± √(b² − 4ac)) / 2a. Compute the discriminant first: its sign tells you whether the roots are real or complex.
What is the discriminant?
D = b² − 4ac. D > 0 gives two distinct real roots, D = 0 gives one repeated root, and D < 0 gives a complex conjugate pair.
What if a = 0?
The equation collapses to bx + c = 0, a linear equation with the single root x = −c/b. If b is also 0 there are either no roots or infinitely many.
How do repeated roots work?
When D = 0 both formula branches coincide at x = −b/2a. The parabola just touches the x-axis at its vertex (a double root).
What are complex roots of a quadratic?
When D < 0 the roots are −b/2a ± i·√|D|/2a. They come in conjugate pairs and the parabola never crosses the x-axis.
How do I verify quadratic roots?
Substitute each root back: ax² + bx + c must equal 0. Equivalently check Vieta: roots sum to −b/a and multiply to c/a.
What is the vertex connection?
The vertex x-coordinate −b/2a is the midpoint of the two roots (real or complex). The axis of symmetry passes through it.
Can quadratics be solved by factoring?
Sometimes: if roots r₁, r₂ are rational, ax² + bx + c = a(x − r₁)(x − r₂). The formula always works, factoring only when the numbers are friendly.
Where are quadratic roots used?
Projectile motion, profit break-even points, optics focal calculations, circuit resonance, and anywhere a squared relationship appears.

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Glossary

Technical terms used in mortgage calculations explained in plain language.

Root (zero)
An x-value making the polynomial equal zero.
Discriminant
D = b² − 4ac; classifies root type.
Quadratic formula
x = (−b ± √D)/2a, solving any quadratic.
Double root
A repeated root when D = 0.
Complex conjugate
Pair a ± bi arising when D < 0.
Vertex
The parabola turning point at x = −b/2a.
Vieta's formulas
Root sum −b/a and product c/a.
Axis of symmetry
Vertical line x = −b/2a through the vertex.

Authoritative References

Trusted sources and standards referenced by this calculator.

Applicable Standards

Common Core A-REI — solve quadratic equationsNCTM Algebra — functions and complex numbers

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Page Information

Last Updated:

2026-09-29

Version:

2.0.0

License:

Free to use for educational and professional purposes.