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Completing the square is a method used to transform any quadratic equation into a perfect square trinomial. This technique helps simplify the process of solving for variables, and is particularly useful in turning equations into a form that makes graphing parabolas easier. Here's how it generally works:

• First, ensure the quadratic and linear terms are on one side of the equation. In this example, we rearranged the terms as \( x = 20y^2 - 40y + 25 \).
• Next, factor out the coefficient of the quadratic term (here, 20) from the terms involving \( y \). This gives us:\( x = 20(y^2 - 2y) + 25 \).
• Find the value needed to complete the square: Take half of the linear coefficient (\(-2\)), square it (\((-2/2)^2 = 1\)), and add and subtract it within the parentheses: \( x = 20((y^2 - 2y + 1) - 1) + 25 \).
• Simplify to the completed square form:\( x = 20(y - 1)^2 - 5 \).

Completing the square transforms the equation into a format that reveals important features, such as the vertex, and helps with further analysis and graphing of the parabola.

The vertex of a parabola is a key component that represents the turning point of the graph. It is either the highest or lowest point, depending on the orientation of the parabola.In the standard form equation \((y - k)^2 = 4p(x - h)\), the vertex of the parabola is directly found at the point \((h, k)\).Given the equation after completing the square, \( x = 20(y - 1)^2 - 5 \),we can see:

• \( h = 5 \)
• \( k = 1 \)

Thus, the vertex is at \((5, 1)\).This vertex tells us where the parabola is centered on the graph and is essential for sketching and understanding the graph's structure.

The focus and directrix are fundamental elements in defining the shape and orientation of a parabola. These components are linked by the concept that a parabola consists of all points equidistant from a point (focus) and a line (directrix).Once we have our parabolic equation in the standard form, we identify the location of these elements:

• The given completed square equation, \( x = 20(y - 1)^2 - 5 \), can be rewritten as \((y - 1)^2 = 4 \cdot 5 \cdot (x - 5)\),indicating \( 4p = 20 \), leading us to \( p = 5 \).
• The focus is found at \((h + p, k) = (10, 1)\), meaning it is located to the right of the vertex, indicating a rightward opening parabola.
• The directrix, a vertical line at \( x = h - p = 0 \), is positioned to the left of the vertex.

Understanding these points assists with accurately sketching the parabola's path and confirming its orientation.

The equation of a parabola can take various forms, but commonly it is expressed in the vertex form as \((y - k)^2 = 4p(x - h)\). This format highlights the significant features of a parabola, such as its vertex, focus, and directrix, guiding us in graphing and analyzing the curve.In the given exercise, we start with a quadratic equation and through completing the square and adjusting terms, it is rewritten into a form that easily gives insight into the parabola's key features.

• The original equation \( x = 20y^2 - 40y + 25 \),when rewritten as \((y - 1)^2 = 4 \cdot 5 \cdot (x - 5)\), signifies a parabolic graph where:
• \( h = 5 \) and \( k = 1 \), giving the vertex \((5, 1)\).
• \( p = 5 \), confirming the primary distance used in locating the focus and directrix.

By converting to this clear vertex form, the relevant features that define the parabola become much easier to locate and analyze. Understanding this transformation is crucial for graphing and comprehending parabolic equations.