To find the length [tex]\( x \)[/tex] of the rectangular vegetable garden, we start with the given equation that relates the length and width of the garden to its area:
[tex]\[
x(x - 3) = 54
\][/tex]
1. First, let's expand and rewrite the equation:
[tex]\[
x^2 - 3x = 54
\][/tex]
2. Next, we'll move all terms to one side to set the equation to zero:
[tex]\[
x^2 - 3x - 54 = 0
\][/tex]
3. Now we solve this quadratic equation. We need to find the roots of the equation [tex]\( x^2 - 3x - 54 = 0 \)[/tex]. To do this, we can factorize the quadratic equation:
We look for two numbers that multiply to [tex]\(-54\)[/tex] (the constant term) and add to [tex]\(-3\)[/tex] (the coefficient of [tex]\( x \)[/tex]). Those numbers are [tex]\( -9 \)[/tex] and [tex]\( 6 \)[/tex].
4. Using these numbers, we can write:
[tex]\[
x^2 - 9x + 6x - 54 = 0
\][/tex]
5. Factor by grouping:
[tex]\[
x(x - 9) + 6(x - 9) = 0
\][/tex]
6. Factor out the common term [tex]\((x - 9)\)[/tex]:
[tex]\[
(x - 9)(x + 6) = 0
\][/tex]
7. Set each factor equal to zero to solve for [tex]\( x \)[/tex]:
[tex]\[
x - 9 = 0 \quad \text{or} \quad x + 6 = 0
\][/tex]
[tex]\[
x = 9 \quad \text{or} \quad x = -6
\][/tex]
Since the length of the garden must be a positive value, we discard [tex]\( x = -6 \)[/tex] as it is not physically meaningful in this context.
Thus, the length [tex]\( x \)[/tex] of the garden is [tex]\( 9 \)[/tex] feet.
The length is [tex]\(\boxed{9}\)[/tex] feet.
