Lila is planning to build a rectangular garden in her backyard. She wants the length to be twice the width plus 5 meters. If the area of her garden is 250 square meters, then what is the width?(1 point)

Question

meters

Answer 1

Let's denote the width of the garden as \( w \) meters. According to the problem, the length \( l \) can be expressed as:

\[ l = 2w + 5 \]

The area \( A \) of the rectangular garden is given by the formula:

\[ A = l \cdot w \]

Substituting the expression for length into the area formula gives:

\[ 250 = (2w + 5)w \]

Expanding this equation:

\[ 250 = 2w^2 + 5w \]

Rearranging the equation to set it to 0:

\[ 2w^2 + 5w - 250 = 0 \]

Now, we can apply the quadratic formula to solve for \( w \). The quadratic formula is:

\[ w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Here, \( a = 2 \), \( b = 5 \), and \( c = -250 \). First, we calculate the discriminant \( b^2 - 4ac \):

\[ b^2 - 4ac = 5^2 - 4 \cdot 2 \cdot (-250) = 25 + 2000 = 2025 \]

Now we find the square root of the discriminant:

\[ \sqrt{2025} = 45 \]

Substituting back into the quadratic formula:

\[ w = \frac{-5 \pm 45}{2 \cdot 2} = \frac{-5 \pm 45}{4} \]

Calculating the two potential solutions for \( w \):

1. Using the positive solution:

\[ w = \frac{40}{4} = 10 \]

2. Using the negative solution:

\[ w = \frac{-50}{4} = -12.5 \quad \text{(not valid as width cannot be negative)} \]

Thus, the valid width of the garden is:

\[ \boxed{10} \text{ meters} \]