To solve the problem, let's start by understanding the given information:
1. The width of the rectangular garden is 2 feet less than the length.
2. The area of the garden is 48 square feet.
3. The length of the garden is represented by [tex]\( x \)[/tex].
The equation provided is based on the relationship between the length, width, and area:
[tex]\[ x(x - 2) = 48 \][/tex]
Let's break down the steps to solve for [tex]\( x \)[/tex]:
1. Start with the equation:
[tex]\[ x(x - 2) = 48 \][/tex]
2. Expand and rearrange the equation to set it to zero:
[tex]\[ x^2 - 2x = 48 \][/tex]
[tex]\[ x^2 - 2x - 48 = 0 \][/tex]
This is a quadratic equation in the standard form:
[tex]\[ ax^2 + bx + c = 0 \][/tex]
where [tex]\( a = 1 \)[/tex], [tex]\( b = -2 \)[/tex], and [tex]\( c = -48 \)[/tex].
3. Solve the quadratic equation using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Substitute [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] into the formula:
[tex]\[ x = \frac{2 \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot (-48)}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{2 \pm \sqrt{4 + 192}}{2} \][/tex]
[tex]\[ x = \frac{2 \pm \sqrt{196}}{2} \][/tex]
[tex]\[ x = \frac{2 \pm 14}{2} \][/tex]
4. Calculate the possible solutions for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{2 + 14}{2} = \frac{16}{2} = 8 \][/tex]
[tex]\[ x = \frac{2 - 14}{2} = \frac{-12}{2} = -6 \][/tex]
The possible solutions are [tex]\( x = 8 \)[/tex] and [tex]\( x = -6 \)[/tex].
5. Since the length of the garden cannot be negative, we discard [tex]\( x = -6 \)[/tex].
Therefore, the length of the garden, [tex]\( x \)[/tex], is [tex]\( 8 \)[/tex] feet.
Thus, the length of the garden is [tex]\(\boxed{8}\)[/tex] feet.
