%5E2%2B(x-11)%5E2.jpeg "(x-3)^2+(x-11)^2")
Exploring the Expression (x-3)^2 + (x-11)^2
This expression represents the sum of squares of two linear terms: (x-3) and (x-11). It appears in various mathematical contexts, particularly when dealing with distance and geometry. Let's delve deeper into its properties and potential applications.
Understanding the Basics
-
Expansion: We can expand the expression using the FOIL method (First, Outer, Inner, Last) for squaring binomials:
(x-3)^2 + (x-11)^2 = (x^2 - 6x + 9) + (x^2 - 22x + 121)
Simplifying, we get: 2x^2 - 28x + 130
-
Graphical Representation: The expression represents the equation of a parabola when set equal to a constant. The parabola opens upwards due to the positive coefficient of the x^2 term.
-
Minimum Value: The expression has a minimum value at the vertex of the parabola. We can find this minimum using the formula for the x-coordinate of the vertex:
x = -b / 2a = 28 / (2 * 2) = 7
Substituting x = 7 back into the original expression, we find the minimum value:
(7 - 3)^2 + (7 - 11)^2 = 16 + 16 = 32
Applications
-
Distance Formula: The expression is closely related to the distance formula. If we consider (3,0) and (11,0) as points on the x-axis, then the expression represents the square of the distance between a point (x,0) and these two fixed points.
-
Optimization: In optimization problems, the expression might represent a cost function or a function to be minimized. Finding its minimum value is crucial for identifying the optimal solution.
-
Quadratic Equations: Setting the expression equal to a constant leads to a quadratic equation. Solving this equation provides the x-values where the parabola intersects a given horizontal line.
Summary
(x-3)^2 + (x-11)^2 is a simple yet powerful expression that embodies various mathematical concepts. Understanding its expansion, graphical representation, and applications can be valuable in solving problems involving distance, optimization, and quadratic equations.