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Completing the square is a method used in algebra to transform a quadratic expression into a perfect square trinomial, which makes it easier to analyze and solve. In the context of contour lines given by the equation \(z = x^2 + y^2 - 2x - 2y\), completing the square helps us identify the geometric shapes described by the equation.
To complete the square for the \(x\) terms, you focus on \(x^2 - 2x\). For this expression:

• Take the coefficient of \(x\), which is \(-2\).
• Divide it by 2 to get \(-1\).
• Square the result to get \(1\).
• Add and subtract this square within the expression: \(x^2 - 2x = (x-1)^2 - 1\).

A similar process is applied to the \(y\) terms \(y^2 - 2y\), producing \((y-1)^2 - 1\). By completing the squares, we naturally transform the expression into \((x-1)^2 + (y-1)^2 - 2 = c\). This step redefines our initial expression into a form that's easier to manipulate and understand geometrically.

The process of completing the square reveals the contour equation as \((x-1)^2 + (y-1)^2 = c + 2\). This equation is the standard form of a circle in a plane.
By comparing with the general equation \((x-h)^2 + (y-k)^2 = r^2\), we see:

• \((h, k) = (1, 1)\) is the circle's center.
• \(r^2 = c + 2\).
• The radius \(r\) is \(\sqrt{c + 2}\).

The transformation illustrates how geometric figures, like circles, can emerge from algebraic manipulation. The circle's radius depends directly on the variable \(c\), showing that each distinct \(c\) gives a circle with a corresponding radius.

To understand how different values of \(c\) affect the contour lines, you need to evaluate the relationship \((x-1)^2 + (y-1)^2 = c + 2\).
Here's how different \(c\) values influence the contours:

• If \(c = -2\), then \(r^2 = 0\). Thus, the circle degenerates into the single point \((1, 1)\).
• If \(c > -2\), the radius \(r\) is real and positive, leading to circles centered at \((1, 1)\).
• If \(c < -2\), the term \(c + 2\) becomes negative, yielding an imaginary radius. Therefore, no real circle exists in this scenario.

This analysis is vital for visualizing potential contour lines and understanding limitations based on \(c\). Each variation of \(c\) showcases a different instance of how the function's value affects the geometric configuration in a two-dimensional plane.