1. **State the problem:** We have a flower bed with dimensions width $x$ meters and length $3x$ meters. There is a walkway 9 meters wide surrounding the flower bed on all sides. The combined area of the flower bed and walkway is 399 square meters. 2. **Write the expression for the total area:** The total length including the walkway is $x + 2 \times 9 = x + 18$ meters. The total width including the walkway is $3x + 2 \times 9 = 3x + 18$ meters. 3. **Write the quadratic equation:** The total area is given by $$ (x + 18)(3x + 18) = 399 $$ 4. **Expand the equation:** $$ 3x^2 + 18x + 54x + 324 = 399 $$ $$ 3x^2 + 72x + 324 = 399 $$ 5. **Simplify by subtracting 399 from both sides:** $$ 3x^2 + 72x + 324 - 399 = 0 $$ $$ 3x^2 + 72x - 75 = 0 $$ 6. **Divide the entire equation by 3 to simplify:** $$ \cancel{3}x^2 + \cancel{3}24x - \cancel{3}25 = 0 $$ $$ x^2 + 24x - 25 = 0 $$ 7. **Solve the quadratic equation using the quadratic formula:** $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$ where $a=1$, $b=24$, and $c=-25$. Calculate the discriminant: $$ \Delta = 24^2 - 4 \times 1 \times (-25) = 576 + 100 = 676 $$ Calculate the roots: $$ x = \frac{-24 \pm \sqrt{676}}{2} = \frac{-24 \pm 26}{2} $$ Two possible solutions: $$ x = \frac{-24 + 26}{2} = \frac{2}{2} = 1 $$ $$ x = \frac{-24 - 26}{2} = \frac{-50}{2} = -25 $$ Since $x$ represents a length, it must be positive, so $x = 1$ meter. 8. **Find the dimensions of the flower bed:** Width = $x = 1$ meter Length = $3x = 3 \times 1 = 3$ meters **Final answer:** Length of flower bed: 3 meter(s) Width of flower bed: 1 meter(s)