Real Roots Calculator Mathway

This real roots calculator helps you find the real solutions to polynomial equations. Whether you're solving quadratic, cubic, or quartic equations, this tool provides accurate results and explains the process step-by-step.

What Are Real Roots?

Real roots, also known as real solutions, are the values of x that satisfy a polynomial equation and result in a real number. For example, in the equation x² - 5x + 6 = 0, the real roots are x = 2 and x = 3.

Real roots are distinct from complex roots, which involve imaginary numbers. The number of real roots a polynomial has depends on its degree and the nature of its coefficients.

How to Find Real Roots

Finding real roots involves solving polynomial equations. Here are the common methods:

Factoring: Express the polynomial as a product of simpler polynomials and solve for x.
Quadratic Formula: For quadratic equations (degree 2), use the formula x = [-b ± √(b² - 4ac)] / (2a).
Numerical Methods: For higher-degree polynomials, use methods like the Newton-Raphson method or graphing to approximate real roots.

Our real roots calculator uses a combination of these methods to provide accurate results for polynomials up to degree 4.

Real Roots Formula

Quadratic Formula

For a quadratic equation ax² + bx + c = 0, the real roots are given by:

x = [-b ± √(b² - 4ac)] / (2a)

The discriminant (b² - 4ac) determines the nature of the roots:

If discriminant > 0: Two distinct real roots
If discriminant = 0: One real root (repeated)
If discriminant < 0: No real roots (complex roots)

Note

For polynomials of degree 3 or higher, exact formulas for real roots become more complex and may not always yield simple expressions. In such cases, numerical methods or graphing are often used.

Example Calculation

Let's solve the quadratic equation x² - 5x + 6 = 0 using the quadratic formula.

Identify coefficients: a = 1, b = -5, c = 6
Calculate discriminant: b² - 4ac = (-5)² - 4(1)(6) = 25 - 24 = 1
Apply quadratic formula: x = [5 ± √1] / 2
Find roots: x = (5 + 1)/2 = 3 and x = (5 - 1)/2 = 2

The real roots of the equation are x = 2 and x = 3.

FAQ

What is the difference between real and complex roots?

Real roots are solutions that are real numbers, while complex roots involve imaginary numbers. For example, the equation x² + 1 = 0 has complex roots x = ±i, where i is the imaginary unit.

Can all polynomials have real roots?

No, not all polynomials have real roots. For example, the equation x² + 1 = 0 has no real roots. The number of real roots depends on the polynomial's degree and coefficients.

How do I know if a polynomial has real roots?

You can use the discriminant for quadratic equations. For higher-degree polynomials, you can use graphing or numerical methods to check for real roots.