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Completing the square is a method used to transform a quadratic expression into a perfect square trinomial. This is particularly useful in finding the vertex form of a parabola, which makes it easier to identify key features like the vertex.

For the equation \[x = -3y^2 + 30y\]we'll start by focusing on the part inside the parentheses when \[x = -3(y^2 - 10y)\].
To complete the square:

• Take half of the linear coefficient, which in this case is -10, giving us -5.
• Square it: \((-5)^2 = 25\). This is the number we'll use to complete the square.

Rewrite the quadratic expression as a perfect square trinomial:\[y^2 - 10y = (y - 5)^2 - 25\]
This transformation' will help us rewrite the original equation into vertex form, aiding in graphing and analysis.

The vertex of a parabola is the point where it turns around; it's the highest or lowest point depending on the parabola’s direction. For our parabola, using the completed square, we rewrite the equation\[x = -3(y - 5)^2 + 75\]This equation is now in vertex form:

• Here, \((y - 5)^2\) represents the squared term.
• The number 5 is the y-component of the vertex, \(k = 5\).
• The term 75 is the x-component of the vertex, \(h = 75\).

This means the vertex of this parabola is at the point \((75, 5)\).
Understanding the vertex helps in plotting the graph of the parabola efficiently.

The equation of a parabola can be expressed in different forms, and understanding these can greatly simplify the graphing and analysis process. In the context of our exercise, we began with a standard quadratic form:\[x = -3y^2 + 30y\]Translating this into vertex form, as we completed the square, gave us:\[x = -3(y - 5)^2 + 75\]Vertex form is highly beneficial because it clearly outlines the vertex of the parabola. By looking at \[x = a(y-k)^2 + h\], where \(a\) indicates the direction and width of the parabola, and \((h, k)\) represents the vertex.

• Here, \(a = -3\), indicating the parabola opens to the left because it is negative.
• The values \(h = 75\) and \(k = 5\)form the vertex.

Recognizing this format helps sketch the parabola accurately and illuminate the behavior of the quadratic function visually on a graph.