what is a double root

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26-12-2024

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Understanding Double Roots: When Solutions Repeat

In the world of mathematics, specifically algebra, we often encounter equations that need solving. These equations might be simple or complex, linear or quadratic, but the solutions, or roots, represent the values that make the equation true. Sometimes, however, we find a special kind of solution: the double root. But what exactly is a double root, and how does it differ from other solutions?

A double root, also known as a repeated root or root of multiplicity two, is a solution to an equation that appears twice. This means the equation's graph touches the x-axis at a single point instead of crossing it. Let's break this down further with examples.

Understanding through Quadratic Equations:

Quadratic equations, those of the form ax² + bx + c = 0, are a common place to encounter double roots. The solutions to a quadratic equation can be found using the quadratic formula:

x = [-b ± √(b² - 4ac)] / 2a

The term inside the square root, (b² - 4ac), is called the discriminant. The discriminant determines the nature of the roots:

• If (b² - 4ac) > 0: There are two distinct real roots.
• If (b² - 4ac) = 0: There is one real root (a double root).
• If (b² - 4ac) < 0: There are two complex roots (involving imaginary numbers).

Example of a Double Root:

Consider the quadratic equation x² - 4x + 4 = 0. Using the quadratic formula:

x = [4 ± √(16 - 16)] / 2 = [4 ± 0] / 2 = 2

In this case, we get x = 2 as the only solution. However, this is a double root because the equation can be factored as (x - 2)(x - 2) = 0, clearly showing the root 2 appearing twice. Graphically, this means the parabola touches the x-axis at x = 2 but doesn't cross it.

Beyond Quadratic Equations:

Double roots aren't limited to quadratic equations. They can occur in higher-degree polynomial equations as well. For example, the equation x³ - 6x² + 12x - 8 = 0 has a triple root at x = 2, meaning the solution 2 appears three times. A double root is just a specific case of a root with multiplicity greater than one.

Graphical Interpretation:

The graphical representation provides a clear visual understanding. For a simple quadratic equation with a double root, the parabola touches the x-axis at the point representing the double root and then turns around without crossing the axis. For higher-degree polynomials, the behavior is more complex, but the point corresponding to the double root will show a similar "touch and turn" behavior.

In Conclusion:

A double root represents a repeated solution to an equation. It's a crucial concept in algebra and calculus, impacting the behavior of functions and their graphs. Understanding the concept of double roots, and more generally roots of higher multiplicity, is essential for a deeper understanding of equation solving and function analysis.