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Roots Equation Calculator

Reviewed by Calculator Editorial Team

A roots equation calculator helps solve quadratic equations of the form ax² + bx + c = 0. These equations are fundamental in algebra and have applications in physics, engineering, and economics. The calculator provides solutions for both real and complex roots, along with visualizations to help understand the results.

What is a Roots Equation?

A roots equation, also known as a quadratic equation, is a second-degree polynomial equation in the form:

ax² + bx + c = 0

Where:

  • a is the coefficient of x² (must not be zero)
  • b is the coefficient of x
  • c is the constant term

The roots of the equation are the values of x that satisfy the equation. These roots can be real or complex numbers, depending on the discriminant (b² - 4ac).

The Quadratic Formula

The standard method for solving quadratic equations is the quadratic formula:

x = [-b ± √(b² - 4ac)] / (2a)

The discriminant (D) determines the nature of the roots:

  • If D > 0: Two distinct real roots
  • If D = 0: One real root (repeated)
  • If D < 0: Two complex conjugate roots

The quadratic formula works for all quadratic equations where a ≠ 0. It's derived from completing the square, a fundamental algebraic technique.

How to Use the Calculator

  1. Enter the coefficients a, b, and c in the input fields
  2. Click the "Calculate" button to solve the equation
  3. View the results including the roots and discriminant
  4. Use the chart to visualize the quadratic function
  5. Click "Reset" to clear the inputs and results

Example Input

For the equation x² - 5x + 6 = 0, enter:

  • a = 1
  • b = -5
  • c = 6

Example Calculation

Let's solve x² - 5x + 6 = 0 using the calculator:

  1. Calculate the discriminant: D = (-5)² - 4(1)(6) = 25 - 24 = 1
  2. Since D > 0, there are two real roots
  3. Apply the quadratic formula:
    • x₁ = [5 + √1]/2 = 3
    • x₂ = [5 - √1]/2 = 2

The roots are x = 2 and x = 3. The calculator will display these results along with a graph of the quadratic function.

Frequently Asked Questions

What is the difference between real and complex roots?

Real roots are actual numbers that satisfy the equation, while complex roots involve imaginary numbers (√-1). The discriminant determines whether roots are real or complex.

Can the quadratic formula solve any quadratic equation?

Yes, the quadratic formula can solve any quadratic equation as long as the coefficient of x² (a) is not zero. It's a universal method for solving these equations.

What does a negative discriminant mean?

A negative discriminant indicates that the quadratic equation has two complex conjugate roots. These roots are complex numbers that are conjugates of each other.

How can I verify the roots I found?

You can verify the roots by substituting them back into the original equation. If both sides of the equation are equal, the roots are correct.