Factor the polynomial.
$x^2-6 x+9$
To factor the quadratic polynomial \(x^2 - 6x + 9\), we look for two numbers whose product is the constant term, \(9\), and whose sum is the coefficient of the \(x\)-term, \(-6\). Let these numbers be \(p\) and \(q\). We need:
\[p \cdot q = 9\]
\[p + q = -6\]
By testing pairs of factors of \(9\), we see that \(-3\) and \(-3\) satisfy these conditions, as \((-3) \cdot (-3) = 9\) and \((-3) + (-3) = -6\).
The polynomial \(x^2 - 6x + 9\) can be written in the form \((x + p)(x + q)\). Substituting \(p = -3\) and \(q = -3\), we get:
\[x^2 - 6x + 9 = (x - 3)(x - 3)\]
Alternatively, we can recognize this as a perfect square trinomial of the form \(a^2 - 2ab + b^2 = (a - b)^2\). Here, \(a = x\) and \(b = 3\), so:
\[x^2 - 2(x)(3) + 3^2 = (x - 3)^2 = (x - 3)(x - 3)\]
Comparing this result to the given options, it matches option D.
In architecture and structural design, calculating the area of a square courtyard involves squaring a binomial expression. For instance, if the side length of a square garden is given as \((x - 3)\) meters, the total area is calculated as \((x - 3)^2 = x^2 - 6x + 9\) square meters. Factoring expressions like this allows engineers and designers to work backwards from a known area to determine the dimensions of a space, facilitating efficient planning and resource allocation.