%5E2%2B(x-2)%5E2%2B(x-3)%5E2.jpeg "(x-1)^2+(x-2)^2+(x-3)^2")
Exploring the Expression (x-1)² + (x-2)² + (x-3)²
This expression represents the sum of the squares of three consecutive differences. Let's delve deeper into its properties and applications.
Understanding the Expression
At its core, the expression represents a simple algebraic manipulation. Each term is a squared difference between the variable x and a constant: 1, 2, and 3. This structure gives us a hint about potential applications in geometric or optimization contexts.
Expanding and Simplifying
We can expand the expression by using the binomial theorem or by direct multiplication:
(x-1)² + (x-2)² + (x-3)² = (x² - 2x + 1) + (x² - 4x + 4) + (x² - 6x + 9)
Combining like terms, we get:
(x-1)² + (x-2)² + (x-3)² = 3x² - 12x + 14
Finding the Minimum Value
The simplified form of the expression, 3x² - 12x + 14, is a quadratic function. We know that quadratic functions have a minimum or maximum value. To find the minimum value of this expression, we can use the vertex formula:
x = -b / 2a
where a and b are the coefficients of the quadratic equation. In this case, a = 3 and b = -12.
Therefore, the minimum value of the expression is achieved at:
x = -(-12) / (2 * 3) = 2
To find the actual minimum value, we substitute x = 2 back into the expression:
3(2)² - 12(2) + 14 = 2
Geometric Interpretation
The expression (x-1)² + (x-2)² + (x-3)² can be interpreted geometrically. Imagine points A(1,0), B(2,0), and C(3,0) on the x-axis. If we take any point P(x,0) on the x-axis, then the expression represents the sum of the squares of the distances between P and A, P and B, and P and C.
In other words, the expression gives us the squared distance between point P and the centroid of the triangle formed by points A, B, and C. This connection helps us visualize the minimum value obtained earlier.
Applications
This expression can be used in various applications:
- Optimization: Minimizing the sum of squares of deviations is a common problem in optimization. This expression could be used to find the optimal value of x that minimizes the sum of squared differences between x and three given values.
- Statistics: In statistics, the expression is similar to the formula for variance. It represents the sum of squared deviations from the mean.
- Geometric Modeling: The geometric interpretation of the expression could be used in problems involving finding the point on a line closest to three given points.
Conclusion
The expression (x-1)² + (x-2)² + (x-3)² is a simple yet versatile algebraic construct. By understanding its expansion, simplification, and geometric interpretation, we can explore its applications in various fields, from optimization to geometry and statistics.