Answer
$x = 5$ meters
Work Step by Step
Since the exercise says that the Area of both the pool and the path combined measures 600 square meters, we can write the following equation: $$Area = 600 = (10 + 2x)(20 + 2x)$$ in accordance with the figure in the exercise. By using the FOIL method, we can expand the exercise as follows: $$600 = 10(20) + 10(2x) + 2x(20) + 2x(2x)$$ $$600 = 200 + 20x + 40x + 4x^2$$ $$600 = 4x^2 + 60x + 200$$ By rearranging the equation: $$0 = 4x^2 + 60x + 200 - 600$$ $$0 = 4x^2 + 60x - 400$$ which is a quadratic equation. Now, we can solve for $x$ using the Quadratic Formula $x=\frac{-b\frac{+}{} \sqrt{b^2 - 4ac}}{2a}$:
$$x=\frac{-(60)\frac{+}{} \sqrt{(60)^2 - 4(4)(-400)}}{2(4)}$$ $$x=\frac{-60\frac{+}{} \sqrt{3600 + 6,400}}{8}$$ $$x=\frac{-60\frac{+}{} \sqrt{10,000}}{8}$$ $$x=\frac{-60\frac{+}{} \sqrt{100 \times 100}}{8}$$ $$x=\frac{-60\frac{+}{} 100}{8}$$ $$x = 5$$ $$or$$ $$x = -20$$ Since we're dealing with measurements, negative values make no sense, so the only viable solution is $x = 5$ meters.
