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Square Equation Root Calculator

Reviewed by Calculator Editorial Team

A square equation, also known as a quadratic equation, is a second-degree polynomial equation in a single variable. The general form is ax² + bx + c = 0, where a, b, and c are constants, and x represents the variable. Solving for x yields the roots of the equation.

What is a Square Equation?

A square equation is a type of quadratic equation that can be written in the standard form:

ax² + bx + c = 0

Where:

  • a, b, and c are constants
  • x is the variable
  • a ≠ 0 (if a = 0, the equation becomes linear)

The solutions to this equation are called roots or zeros. A quadratic equation can have:

  • Two distinct real roots
  • One real root (a repeated root)
  • No real roots (the roots are complex numbers)

The Quadratic Formula

The most common method for solving quadratic equations is the quadratic formula:

x = [-b ± √(b² - 4ac)] / (2a)

Where:

  • √(b² - 4ac) is called the discriminant
  • The discriminant determines the nature of the roots:
Discriminant Nature of Roots
b² - 4ac > 0 Two distinct real roots
b² - 4ac = 0 One real root (repeated)
b² - 4ac < 0 No real roots (complex roots)

The quadratic formula works for all quadratic equations except when a = 0, which would make it a linear equation.

How to Use This Calculator

  1. Enter the coefficients a, b, and c in the input fields
  2. Click the "Calculate" button
  3. View the results including the roots and discriminant
  4. Review the explanation of the calculation
  5. Use the chart to visualize the quadratic function

Note: This calculator assumes real coefficients. For complex coefficients, additional mathematical techniques are required.

Example Calculation

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

  1. Identify the coefficients: a = 1, b = -5, c = 6
  2. Calculate the discriminant: (-5)² - 4(1)(6) = 25 - 24 = 1
  3. Apply the quadratic formula:

    x = [5 ± √1] / 2

  4. Calculate the two roots:
    • x₁ = (5 + 1)/2 = 3
    • x₂ = (5 - 1)/2 = 2

The roots of the equation x² - 5x + 6 = 0 are x = 2 and x = 3.

Frequently Asked Questions

What is the difference between a linear and quadratic equation?

A linear equation has a single variable with the highest power of 1, while a quadratic equation has a variable with the highest power of 2. The general forms are ax + b = 0 and ax² + bx + c = 0, respectively.

How do I know if a quadratic equation has real roots?

A quadratic equation has real roots if the discriminant (b² - 4ac) is greater than or equal to zero. If the discriminant is negative, the roots are complex numbers.

What is the vertex of a quadratic function?

The vertex of a quadratic function is the point where the parabola changes direction. It can be found using the formula x = -b/(2a). The y-coordinate is then found by substituting x back into the equation.