The problem presented involves finding how many meters of wire are needed to enclose a vegetable garden. The given conditions are as follows:

• The area of the vegetable garden is 80 m².
• The shape of the garden has one side marked as $x + 2$ and another side marked as $x$.
• A 1-meter wide iron net fence is to be built around the perimeter of the garden.

Approach:

1. Calculate dimensions:

   • The area $A$ of the garden is given by the formula for the area of a rectangle, which is: $A = \text{Length} \times \text{Width}$
   • The length and width are $x + 2$ and $x$, respectively. Hence: $A = (x + 2) \times x = 80 \, \text{m}^2$ Expanding the equation: $x^2 + 2x = 80$ Solving the quadratic equation: $x^2 + 2x - 80 = 0$

2. Solve the quadratic equation: We can solve $x^2 + 2x - 80 = 0$ using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a = 1$, $b = 2$, and $c = -80$.

3. Find the perimeter: Once $x$ is determined, we can calculate the perimeter $P$ of the rectangle: $P = 2 \times (\text{Length} + \text{Width}) = 2 \times ((x + 2) + x)$

Let's proceed with solving the quadratic equation and find the perimeter of the garden.The value of $x$ is 8 meters, which means the dimensions of the garden are:

• Width: $x = 8 \, \text{m}$
• Length: $x + 2 = 10 \, \text{m}$

The total perimeter of the garden is: $P = 36 \, \text{meters}$

Thus, 36 meters of wire are required to enclose the garden.

Would you like more details on any part of the solution or have any further questions?

Here are 5 related questions to expand your understanding:

1. How do we derive the quadratic formula for solving quadratic equations?
2. What would happen if the area of the garden was increased by 20%?
3. How do you find the perimeter for other shapes, like a circle or triangle?
4. Can we solve this problem graphically by plotting the quadratic equation?
5. How do the properties of quadratic functions help in real-life situations like this?

Tip: Always check the discriminant in a quadratic equation to determine if real solutions exist before applying the quadratic formula.