Question $\newline$ Watch Video $\newline$ Show Examples $\newline$ As seen in the diagram below, Xavier is building a walkway with a width of $x$ feet to go around a swimming pool that measures $8$ feet by $6$ feet. If the total area of the pool and the walkway will be $528$ square feet, how wide should the walkway be?

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As seen in the diagram below, Xavier is building a walkway with a width of

x

feet to go around a swimming pool that measures

8

feet by

6

feet. If the total area of the pool and the walkway will be

528

square feet, how wide should the walkway be?

Identify Dimensions and Area: Identify the dimensions of the swimming pool and the total area including the walkway. $\newline$ The swimming pool measures $8$ feet by $6$ feet, and the total area of the pool and walkway is $528$ square feet.
Express Total Area in Terms of $x$ : Express the total area of the pool and walkway in terms of $x$ . The area of the pool is $8 \times 6 = 48$ square feet. The walkway goes around the pool, so it will add $x$ feet to each dimension of the pool. The new dimensions including the walkway will be $(8 + 2x)$ by $(6 + 2x)$ . The total area is therefore $(8 + 2x)(6 + 2x)$ .
Set Up Equation for x: Set up the equation to solve for x. $\newline$ The equation for the total area including the walkway is $(8 + 2x)(6 + 2x) = 528$ .
Expand and Simplify Equation: Expand the equation. $\newline$ Expanding the left side of the equation gives us $8\times 6 + 8\times 2x + 6\times 2x + 4x^2 = 528$ . $\newline$ This simplifies to $48 + 16x + 12x + 4x^2 = 528$ .
Combine Terms and Subtract: Combine like terms and subtract $48$ from both sides to set the equation to zero. $\newline$ Combining like terms gives us $4x^2 + 28x + 48 = 528$ . $\newline$ Subtracting $48$ from both sides gives us $4x^2 + 28x = 480$ .
Divide Equation by $4$ : Divide the entire equation by $4$ to simplify. $\newline$ Dividing by $4$ gives us $x^2 + 7x = 120$ .
Solve Quadratic Equation for $x$ : Solve the quadratic equation for $x$ . We can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. In this case, we will try to factor the equation. We are looking for two numbers that multiply to $120$ and add up to $7$ . These numbers are $15$ and $-8$ , but they do not add up to $7$ , so the equation does not factor nicely. We will use the quadratic formula instead.
Apply Quadratic Formula: Apply the quadratic formula. $\newline$ The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ , where $a = 1$ , $b = 7$ , and $c = -120$ . $\newline$ Plugging in these values gives us $x = \frac{-7 \pm \sqrt{7^2 - 4 \cdot 1 \cdot (-120)}}{2 \cdot 1}$ .
Calculate Discriminant and Solve: Calculate the discriminant and solve for $x$ . The discriminant is $7^2 - 4\cdot1\cdot(-120) = 49 + 480 = 529$ . Taking the square root of $529$ gives us $23$ . So, $x = (-7 \pm 23) / 2$ .
Find Two Possible Solutions: Find the two possible solutions for $x$ . The two possible solutions are $x = (-7 + 23) / 2$ and $x = (-7 - 23) / 2$ . Calculating these gives us $x = 16 / 2$ and $x = -30 / 2$ .
Choose Correct Solution: Choose the solution that makes sense in the context of the problem. $\newline$ Since a negative width does not make sense for the walkway, we discard $x = -\frac{30}{2}$ . The width of the walkway is $x = \frac{16}{2} = 8$ feet.