1. **State the problem:** We have a rectangular pool 20 feet wide and 50 feet long, surrounded by a deck of uniform width. The deck's total area is 456 square feet. We need to find the width of the deck. 2. **Set up variables and formula:** Let the width of the deck be $x$ feet. The total dimensions including the deck are: - Width: $20 + 2x$ - Length: $50 + 2x$ The total area including the pool and deck is: $$ (20 + 2x)(50 + 2x) $$ The pool area alone is: $$ 20 \times 50 = 1000 $$ The deck area is the total area minus the pool area: $$ (20 + 2x)(50 + 2x) - 1000 = 456 $$ 3. **Write the equation:** $$ (20 + 2x)(50 + 2x) - 1000 = 456 $$ 4. **Expand the left side:** $$ 20 \times 50 + 20 \times 2x + 2x \times 50 + 2x \times 2x - 1000 = 456 $$ $$ 1000 + 40x + 100x + 4x^2 - 1000 = 456 $$ 5. **Simplify:** $$ 140x + 4x^2 = 456 $$ 6. **Rewrite as a quadratic equation:** $$ 4x^2 + 140x - 456 = 0 $$ 7. **Divide entire equation by 4 to simplify:** $$ \cancel{4}x^2 + \cancel{140}x - \cancel{456} = 0 \Rightarrow x^2 + 35x - 114 = 0 $$ 8. **Use the quadratic formula:** $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$ where $a=1$, $b=35$, and $c=-114$. 9. **Calculate the discriminant:** $$ b^2 - 4ac = 35^2 - 4 \times 1 \times (-114) = 1225 + 456 = 1681 $$ 10. **Calculate the roots:** $$ x = \frac{-35 \pm \sqrt{1681}}{2} = \frac{-35 \pm 41}{2} $$ 11. **Find the two possible values:** - $$ x = \frac{-35 + 41}{2} = \frac{6}{2} = 3 $$ - $$ x = \frac{-35 - 41}{2} = \frac{-76}{2} = -38 $$ 12. **Interpret the solution:** Since width cannot be negative, the deck width is: $$ \boxed{3 \text{ feet}} $$