91Ó°ÊÓ

Completing the square is a technique used to transform a quadratic equation into a perfect square trinomial. This method makes it easier to rewrite quadratic functions in vertex form, making it simpler to analyze their properties. The process involves the following simple steps:

• Take the quadratic expression in the form of \(ax^2 + bx + c\). For our function \(f(x) = x^2 - 6x + 11\), we begin by focusing on the \(x^2 - 6x\) part.
• To create a perfect square, take half of the linear coefficient \(-6\), square it, and add and subtract this value from the quadratic expression. In this case, \(-\frac{6}{2} = -3\), squared results in \(9\).

After adding and subtracting \(9\), the expression becomes \(x^2 - 6x + 9 - 9 + 11\). Rearrange the terms to group the perfect square trinomial, resulting in \((x - 3)^2 + 2\).
This transformation allows us to work with a more familiar structure, particularly when determining the graph's vertex and axis of symmetry.

The vertex form of a quadratic function provides a clear picture of its graph, particularly the vertex, or the point where the parabola changes direction. Once a quadratic is converted using completing the square, the result naturally fits into the vertex form:

\[f(x) = a(x-h)^2 + k\]
The vertex form is handy because it clearly indicates the vertex \((h, k)\) of the parabola. In our example,

• The expression \((x - 3)^2 + 2\) has \(h = 3\) and \(k = 2\).
• The graph of this function opens upwards since the coefficient \(a = 1\) is positive.

The vertex form reveals that the vertex of the graph \(f(x) = (x-3)^2 + 2\) is at the point \((3, 2)\). Knowing the vertex makes it easy to sketch the graph or solve optimization problems where the maximum or minimum value of the function is of interest.

Though the exercise focuses on rewriting a quadratic using completing the square, it's helpful to know alternative methods for solving quadratic equations, such as the quadratic formula. The quadratic formula is a universal method that provides solutions for any quadratic equation, given by:

\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
This formula can be used directly on quadratic equations to find the roots or zeros of the function. In our case, had we been asked to find the x-intercepts, we could apply:

• Given that \(a = 1\), \(b = -6\), and \(c = 11\), plug these values into the quadratic formula.

Usually, if the discriminant \((b^2 - 4ac)\) is non-negative, real solutions exist. This tool is particularly useful for quadratic functions that are not easily factored. But remember, the quadratic formula is not necessary for transforming the equation into vertex form.