Real Roots of An Equation Calculator

What Are Real Roots?

The real roots of an equation are the real numbers that satisfy the equation. For a polynomial equation like ax³ + bx² + cx + d = 0, real roots are the x-values where the polynomial crosses or touches the x-axis.

Real roots can be found using various methods including:

• Factoring
• Quadratic formula
• Numerical methods (like Newton-Raphson)
• Graphical methods

How to Find Real Roots

Step 1: Identify the Equation Type

First, determine if your equation is linear, quadratic, cubic, or higher-order polynomial. The method for finding roots varies by equation type.

Step 2: Apply the Appropriate Method

For quadratic equations (ax² + bx + c = 0), use the quadratic formula:

For cubic equations, consider using Cardano's formula or numerical methods. For higher-order polynomials, factoring or graphical methods may be more practical.

Step 3: Verify the Solutions

Always plug your solutions back into the original equation to ensure they satisfy it. This step helps catch calculation errors.

Example Calculation

Let's find the real roots of the equation x² - 5x + 6 = 0.

Step 1: Identify Coefficients

Here, a = 1, b = -5, and c = 6.

Step 2: Apply the Quadratic Formula

Plugging into the formula:

Step 3: Calculate Solutions

This gives two real roots:

• x = (5 + 1)/2 = 3
• x = (5 - 1)/2 = 2

Step 4: Verification

Plugging x = 3 back into the equation: 9 - 15 + 6 = 0 ✓

Plugging x = 2 back into the equation: 4 - 10 + 6 = 0 ✓

Common Mistakes

When finding real roots, avoid these common errors:

• Assuming all equations have real roots - some polynomials have only complex roots.
• Forgetting to verify solutions by plugging them back into the original equation.
• Using the wrong formula for the equation type (e.g., applying the quadratic formula to a cubic equation).
• Rounding errors in intermediate calculations that affect the final result.