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Roots of A Quadratic Equation Calculator

Reviewed by Calculator Editorial Team

A quadratic equation is a second-degree polynomial equation in a single variable x with three coefficients: a, b, and c. The general form is ax² + bx + c = 0. This calculator finds the roots of any quadratic equation by applying the quadratic formula.

What is a Quadratic Equation?

A quadratic equation is a polynomial equation of degree 2. It has the general form:

ax² + bx + c = 0

Where:

  • a, b, and c are constants
  • a ≠ 0 (if a = 0, the equation is linear, not quadratic)
  • x is the variable

Quadratic equations can represent many real-world situations, such as projectile motion, area problems, and optimization problems. The solutions to a quadratic equation are called roots or solutions.

The Quadratic Formula

The quadratic formula is a standard method for solving quadratic equations. It provides the roots of the equation in terms of its coefficients.

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: Two complex conjugate roots

Note: The quadratic formula works for all quadratic equations where a ≠ 0. It's derived from completing the square, a method of solving quadratic equations by manipulation.

How to Use This Calculator

  1. Enter the coefficients a, b, and c of your quadratic equation
  2. Click the "Calculate" button
  3. View the roots in the result panel
  4. Interpret the results based on the discriminant

The calculator will display:

  • The roots of the equation
  • The discriminant value
  • A visual representation of the quadratic function
  • An explanation of the results

Example Calculation

Let's solve the quadratic equation: 2x² + 4x - 6 = 0

Step-by-Step Solution

  1. Identify coefficients: a = 2, b = 4, c = -6
  2. Calculate discriminant: b² - 4ac = 16 - (8)(-6) = 16 + 48 = 64
  3. Apply quadratic formula:
    x = [-4 ± √64] / 4
  4. Calculate roots:
    • x₁ = (-4 + 8)/4 = 4/4 = 1
    • x₂ = (-4 - 8)/4 = -12/4 = -3

The roots of the equation are x = 1 and x = -3. This means the quadratic equation equals zero at these two points.

Frequently Asked Questions

What is the difference between roots and solutions?
In the context of quadratic equations, "roots" and "solutions" refer to the same thing - the values of x that satisfy the equation. The term "roots" is more commonly used in algebra, while "solutions" is more general.
Can quadratic equations have complex roots?
Yes, when the discriminant is negative (b² - 4ac < 0), the quadratic equation has two complex conjugate roots. These are solutions in the complex number system.
What does it mean if the discriminant is zero?
A discriminant of zero indicates that the quadratic equation has exactly one real root (a repeated root). This occurs when the parabola touches the x-axis at exactly one point.
How can I verify the roots I've calculated?
You can substitute the calculated roots back into the original quadratic equation to verify they satisfy it. For example, if x = 1 is a root of 2x² + 4x - 6 = 0, substituting 1 should give 0.
What are some real-world applications of quadratic equations?
Quadratic equations model many real-world situations, including projectile motion, optimization problems, area calculations, and growth/decay models. They're fundamental in physics, engineering, and economics.