Complex Roots Calculator


With this calculator, you can get all the complex roots of a polynomial or other algebraic expressions. If the algebraic expression is entered correctly, all solutions will be displayed at the bottom.

How to use the complex roots calculator?

Step 1: Enter the polynomial or algebraic expression in the corresponding input box. You must use * to indicate multiplication between variables and coefficients. For example, enter 2*x or 5*x^2, instead of 2x or 5x^2.

Step 2: Click "Solve" to get all the complex roots of the polynomial or algebraic expression.

Step 3: The roots along with the entered expression will be displayed at the bottom.

How to enter polynomials or expressions in the calculator?

We must consider some aspects to enter polynomials correctly. First, it is important to use * to indicate multiplication between variables and coefficients. For example, instead of entering 4x or 6x, we must enter 4*x or 6*x.

Next, we can use the ^ sign to indicate an exponent. That is, to indicate \(x^3\), we have to enter x^3.

Fractions can also be entered normally using the / sign. For example, to input \(\frac{1}{2}*x\), we have to enter 1/2*x.

Finally, the use of parentheses is recommended to indicate the operations correctly. For example, if we want to input \(\frac{1}{2x}\), we would have to enter 1/(2x).

The following are some examples of how to enter polynomials or algebraic expressions:

  • To write \(x^2+2x+5\), enter x^2+2*x+5.
  • To write \(4x^2-2x+7\), enter 4*x^2-2*x+7.
  • To write \(\frac{1}{2}x^3-\frac{1}{2}x^2+2x\), enter 1/2*x^3-1/2*x^2+2*x.

What are complex roots?

Complex roots refer to solutions of polynomials or algebraic expressions that consist of both real numbers and imaginary numbers.

In the case of polynomials, the Fundamental Theorem of Algebra tells us that any polynomial with coefficients that are real numbers can be completely factored using complex numbers.

In the case of quadratic polynomials, the roots are complex when the determinant is negative.

You can learn more about the complex roots of polynomials in our article Complex Roots of a Polynomial with Examples.

How to find complex roots manually?

We can find complex roots of a quadratic equation by using the quadratic formula:

\( x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\)

By solving the quadratic formula, we will get negative numbers below the square root when the polynomial has complex roots. We simply have to use the imaginary number (square root of -1) to solve.

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