Roots of Equation Calculator

Calculator Form

Polynomial Degree

Variable Symbol

Decimal Places

Iteration Tolerance

Coefficient a6

Term: a6x^6

Coefficient a5

Term: a5x^5

Coefficient a4

Term: a4x^4

Coefficient a3

Term: a3x^3

Coefficient a2

Term: a2x^2

Coefficient a1

Term: a1x

Coefficient a0

Term: a0

Example Data Table

Degree	Equation	Coefficients	Expected Roots
2	x² - 5x + 6 = 0	a2 = 1, a1 = -5, a0 = 6	2, 3
3	x³ - 6x² + 11x - 6 = 0	a3 = 1, a2 = -6, a1 = 11, a0 = -6	1, 2, 3
2	x² + 4 = 0	a2 = 1, a1 = 0, a0 = 4	2i, -2i

Formula Used

The general polynomial equation is:

a_nxⁿ + a_n-1x^n-1 + ... + a₁x + a₀ = 0

For a linear equation, the root is x = -b / a.

For a quadratic equation, the roots are found with:

x = (-b ± √(b² - 4ac)) / 2a

For degree three and above, this calculator uses Durand-Kerner iteration:

r_k,new = r_k - P(r_k) / Π(r_k - r_j), where j is not equal to k.

The residual is |P(root)|. It checks how close each answer is to zero.

How to Use This Calculator

Select the polynomial degree from 1 to 6.
Enter the coefficients for the selected degree.
Use zero for missing terms.
Choose decimal places and tolerance if needed.
Press Calculate Roots.
Review root type, near multiplicity, and residual error.
Use CSV or PDF to save the calculated table.

Root Finding for Polynomial Equations

A roots of equation calculator helps you solve polynomial equations without guessing. It accepts coefficients, builds the equation, and returns every root allowed by the chosen degree. Some roots are real numbers. Other roots are complex numbers. Both are useful in algebra, engineering, signals, and numerical modelling.

Why Roots Matter

A root is a value that makes the equation equal zero. Graphically, a real root shows where the curve crosses or touches the horizontal axis. Complex roots do not appear as x axis crossings, but they still complete the algebraic answer. The calculator also shows residual error. A small residual means the computed root fits the original equation well.

Advanced Calculation Method

Linear and quadratic equations have direct formulas. Higher degree equations usually need numerical iteration. This tool uses a simultaneous root search. It starts with several complex guesses around a circle. Then it improves every guess until the polynomial value becomes very small. This method is helpful because it can find real and complex roots together.

Reading the Results

Each result includes the root, type, and residual. Real roots have an imaginary part near zero. Complex roots appear in pairs when all coefficients are real. Repeated roots may look very close together. Increase decimal places when you need more detail. Use the polynomial value column to compare accuracy across roots.

Practical Uses

Students can check homework steps. Teachers can prepare answer keys. Engineers can test characteristic equations. Finance and science users can study polynomial models. The export buttons keep results easy to record. CSV is useful for spreadsheets. PDF is useful for reports and printed notes.

Good Input Practice

Choose the correct degree first. Enter the leading coefficient carefully. It cannot be zero. Use zero for missing terms. For example, enter zero for the x term in x squared minus nine. Keep coefficients reasonable when possible. Very large values can magnify rounding error. After solving, compare the displayed equation and residuals before saving results.

Helpful Study Notes

Use the calculator as a checking partner, not a replacement for learning. Write the equation first. Predict the possible number of roots. Then compare your manual work with the exported result table and listed formulas for practice.

FAQs

What is a root of an equation?

A root is a value that makes the equation equal zero. For example, x = 2 is a root of x - 2 = 0 because substituting 2 gives zero.

Can this calculator find complex roots?

Yes. It displays complex roots using the form a + bi. Complex roots are common when a polynomial has no matching real crossing on the x axis.

What degree can I enter?

This version supports degrees from 1 to 6. Enter the chosen degree first. Then fill the matching coefficients and use zero for terms that are missing.

Why must the leading coefficient be nonzero?

The leading coefficient defines the selected degree. If it is zero, the equation has a lower degree than selected, so the calculator cannot use that setup correctly.

What does residual mean?

Residual is the absolute value of the polynomial after substituting a calculated root. A smaller residual means the answer is closer to an exact root.

What is near multiplicity?

Near multiplicity counts roots that are extremely close together. It helps identify repeated roots, though numerical rounding can affect very close values.

Should I change the tolerance?

The default tolerance works for most inputs. Use a smaller tolerance for stricter iteration. Use more decimal places when you need a more detailed display.

Why are some answers slightly rounded?

Numerical root finding uses approximations. Rounding also depends on the selected decimal places. Check the residual column to judge answer quality.