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Completing the square is a technique used to rewrite quadratic expressions in a specific form, which helps simplify solving and graphing these equations. Given the quadratic function \[ P(x) = -x^2 - 3x + 10, \]we begin by focusing on the quadratic and linear terms. The idea is to form a perfect square trinomial.

Start by factoring out -1 from the quadratic and linear terms, so we have:\[ P(x) = - (x^2 + 3x) + 10. \]The next step is to "complete the square" inside the parenthesis. This involves:

• Taking the coefficient of the linear term, which is 3,
• Dividing it by 2 to get \( \frac{3}{2}, \)
• Squaring it to get \( \left( \frac{3}{2} \right)^2 = \frac{9}{4}. \)

We then add and subtract this square inside the parenthesis:\[ P(x) = - \left( x^2 + 3x + \frac{9}{4} - \frac{9}{4} \right) + 10. \]This creates a perfect square trinomial:\[ P(x) = - \left( (x + \frac{3}{2})^2 - \frac{9}{4} \right) + 10. \]Once in this form, we simplify to prepare for graphing and identifying other key features.

Vertex form of a quadratic function allows us to easily identify key elements of the parabola, such as its vertex. Once we complete the square, our function is in the form:\[ P(x) = a(x-h)^2 + k, \]where \( a \), \( h \), and \( k \) are constants. For the function above, this results in:\[ P(x) = - (x + \frac{3}{2})^2 + \frac{9}{4} + 10. \]After simplifying the constants, we have:\[ P(x) = - (x + \frac{3}{2})^2 + \frac{49}{4}. \]

In this form:

• \( a = -1 \) indicates that the parabola opens downward,
• \( h = -\frac{3}{2} \) and \( k = \frac{49}{4} \) reveal the vertex of the parabola located at \( \left( -\frac{3}{2}, \frac{49}{4} \right) \).

By having the function in vertex form, it becomes straightforward to graph it by plotting the vertex and recognizing its symmetrical properties.

Graphing a parabola involves plotting points to reveal the curve's shape and direction. With the quadratic in vertex form:\[ P(x) = - (x + \frac{3}{2})^2 + \frac{49}{4}, \]we know that the parabola opens downward because \( a = -1. \) The vertex, \( \left( -\frac{3}{2}, \frac{49}{4} \right), \) is the highest point.

To accurately depict the parabola:

• Plot the vertex on the graph.
• Determine additional points using values around the vertex, such as \( x = -1 \) and \( x = -2. \)

For these values:

• When \( x = -1, \) \( P(x) = \frac{47}{4}. \)
• When \( x = -2, \) the calculation similarly yields \( P(x) = \frac{47}{4}. \)

These points help outline the parabola's symmetrical and downward-opening structure. By connecting these points smoothly, you'll have a clear graph of the quadratic function, showing the parabola's full curve.