To find the length [tex]\( x \)[/tex] of the rectangular vegetable garden, we can start by setting up the given equation based on the problem statement. The width of the garden is described as 4 feet less than the length, and the area of the garden is 140 square feet.
Given:
- Length of the garden = [tex]\( x \)[/tex]
- Width of the garden = [tex]\( x - 4 \)[/tex]
- Area of the garden = 140 square feet
The area of a rectangle is given by the product of its length and width. Therefore, we can set up the equation:
[tex]\[
x \cdot (x - 4) = 140
\][/tex]
Next, we expand and simplify this equation:
[tex]\[
x^2 - 4x = 140
\][/tex]
To solve for [tex]\( x \)[/tex], we must rearrange the equation into standard quadratic form:
[tex]\[
x^2 - 4x - 140 = 0
\][/tex]
We now solve this quadratic equation. We factorize [tex]\( x^2 - 4x - 140 \)[/tex], use the quadratic formula, or another suitable method to find the roots of this quadratic equation.
[tex]\[
x^2 - 4x - 140 = 0
\][/tex]
Solving this quadratic equation yields two potential solutions. However, in the context of the problem, we are looking for a positive length since a negative length does not make sense in this scenario.
Upon solving, we find:
[tex]\[
x = 14
\][/tex]
Hence, the length [tex]\( x \)[/tex] of the garden is:
[tex]\[
\boxed{14} \text{ feet}
\][/tex]
