Jackson has 58 m of fencing to build a four-sided fence around a rectangular plot of land. The area of the land is 204 square meters. Solve for the dimensions (length and width) of the field.

To solve for the dimensions of the rectangular plot of land, we need to set up a system of equations based on the given information.

1. Let $l$ be the length and $w$ be the width of the rectangle.

2. The perimeter of the rectangle is given by the equation: $2l + 2w = 58$ Simplifying this, we get: $l + w = 29 \quad \text{(Equation 1)}$

3. The area of the rectangle is given by the equation: $l \times w = 204 \quad \text{(Equation 2)}$

Now, we need to solve these two equations simultaneously.

From Equation 1, we can express $l$ in terms of $w$: $l = 29 - w$

Substitute this expression for $l$ into Equation 2: $(29 - w) \times w = 204$

This simplifies to: $29w - w^2 = 204$

Rearrange this into a standard quadratic equation form: $w^2 - 29w + 204 = 0$

To solve this quadratic equation, we can use the quadratic formula $w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = 1$, $b = -29$, and $c = 204$.

Calculate the discriminant: $b^2 - 4ac = (-29)^2 - 4 \times 1 \times 204 = 841 - 816 = 25$

Now, apply the quadratic formula: $w = \frac{29 \pm \sqrt{25}}{2}$ $w = \frac{29 \pm 5}{2}$

This gives us two possible solutions for $w$: $w = \frac{29 + 5}{2} = 17 \quad \text{or} \quad w = \frac{29 - 5}{2} = 12$

Using these values of $w$ to find the corresponding $l$:

If $w = 17$: $l = 29 - 17 = 12$

If $w = 12$: $l = 29 - 12 = 17$

Therefore, the dimensions of the rectangular plot of land are: $\text{Length} = 17 \, \text{meters}, \, \text{Width} = 12 \, \text{meters}$ or $\text{Length} = 12 \, \text{meters}, \, \text{Width} = 17 \, \text{meters}$

Both sets of dimensions satisfy the given conditions.