Let the width of the rectangular garden be \( w \) meters. According to the problem, the length \( l \) can be expressed in terms of the width as follows:
\[ l = 2w + 5 \]
The area \( A \) of a rectangle is given by the formula:
\[ A = l \times w \]
From the problem, we know the area is 250 square meters:
\[ A = 250 \quad \text{square meters} \]
Substituting the expression for the length into the area formula gives:
\[ w \times (2w + 5) = 250 \]
Now, we can expand this equation:
\[ 2w^2 + 5w = 250 \]
Next, we will rearrange the equation to set it to zero:
\[ 2w^2 + 5w - 250 = 0 \]
Now we can solve this quadratic equation using the quadratic formula:
\[ w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In our case, \( a = 2 \), \( b = 5 \), and \( c = -250 \). First, we calculate the discriminant:
\[ b^2 - 4ac = 5^2 - 4 \times 2 \times (-250) = 25 + 2000 = 2025 \]
Next, we take the square root of the discriminant:
\[ \sqrt{2025} = 45 \]
Now we can apply the quadratic formula:
\[ w = \frac{-5 \pm 45}{2 \times 2} \]
Calculating the two possible values for \( w \):
- For the positive root:
\[ w = \frac{-5 + 45}{4} = \frac{40}{4} = 10 \]
- For the negative root:
\[ w = \frac{-5 - 45}{4} = \frac{-50}{4} = -12.5 \quad \text{(not physically meaningful)} \]
Thus, the only valid solution is:
\[ w = 10 \quad \text{meters} \]
We should verify this solution. If \( w = 10 \):
\[ l = 2(10) + 5 = 20 + 5 = 25 \quad \text{meters} \]
Now, let’s check the area:
\[ A = l \times w = 25 \times 10 = 250 \quad \text{square meters} \]
Since our calculations are correct, we conclude that the width of Lila's garden is:
\[ \boxed{10} \quad \text{meters} \]
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