Evaluating Quadratics

I drew this because pictures help me visualize the problem.

I named the rectangles 1, 2, 3, and 4. The width of the walk is x.

________

| |__3__| | x

| | | |

|1| |2 |

| | | |

| |_____| |

|_|__4__|_|

x

Let x be the width of the walk. The area of the pool would be 800 square feet (20x40). So the area of the walk must be 1500 - 800, or 700 square feet. Now we have to find ways to represent the area of the walk using only these values: 20, 40, x, and the total area, 700.

First, break up the walk into 4 rectangles. In the picture above, I broke the walk up into two long rectangles (#1 and 2) whose lengths are: the length of the pool (40) plus an x for the width of the walk at the top of the pool, plus another x to account for the width of the walk at the bottom of the pool. So their lengths are 40 + 2x. And their widths are x, the width of the walk.

I also have two small rectanges (#3 and 4) whose lengths are the width of the pool (20) and whose widths are x, the width of the walk.

Now that we now the lengths and widths of all four rectangles, we can find their areas (LxW):

Area of rectangle 1: (40 + 2x)(x) = 40x + 2x

Area of rectangle 2: (40 + 2x)(x) = 40x + 2x

Area of rectangle 3: (20)(x) = 20x

Area of rectangle 4: (20)(x) = 20x

As we already found, we know that the area of the entire walk must be 700 sq ft. So we add the areas of all 4 rectangles together and set it equal to 700. Then we can solve for x, the width of the walk.

40x + 2x

4x

4x

*Note: Now that the right hand side of the equation is 0, we can try to factor this quadratic equation.

*Since every number is divisible by 4, we can simplify the problem to:

x

*At this point you can factor or use the quadratic formula. This does factor easily:

(x - 5)(x + 35) = 0

*Split into two problems:

x - 5 = 0 OR x + 35 = 0

x = 5 OR x = -35

**Since the width of anything cannot be negative, we know the answer is x = 5. So the width of the walk is 5 feet all the way around the pool.**

| |__3__| | x

| | | |

|1| |2 |

| | | |

| |_____| |

|_|__4__|_|

x

Let x be the width of the walk. The area of the pool would be 800 square feet (20x40). So the area of the walk must be 1500 - 800, or 700 square feet. Now we have to find ways to represent the area of the walk using only these values: 20, 40, x, and the total area, 700.

First, break up the walk into 4 rectangles. In the picture above, I broke the walk up into two long rectangles (#1 and 2) whose lengths are: the length of the pool (40) plus an x for the width of the walk at the top of the pool, plus another x to account for the width of the walk at the bottom of the pool. So their lengths are 40 + 2x. And their widths are x, the width of the walk.

I also have two small rectanges (#3 and 4) whose lengths are the width of the pool (20) and whose widths are x, the width of the walk.

Now that we now the lengths and widths of all four rectangles, we can find their areas (LxW):

Area of rectangle 1: (40 + 2x)(x) = 40x + 2x

^{2}Area of rectangle 2: (40 + 2x)(x) = 40x + 2x

^{2}Area of rectangle 3: (20)(x) = 20x

Area of rectangle 4: (20)(x) = 20x

As we already found, we know that the area of the entire walk must be 700 sq ft. So we add the areas of all 4 rectangles together and set it equal to 700. Then we can solve for x, the width of the walk.

40x + 2x

^{2}+ 40x + 2x^{2}+ 20x + 20x = 700 *add all areas together, set = to 7004x

^{2}+ 120x = 700 *combine like terms4x

^{2}+ 120x - 700 = 0 *subtract 700 from both sides*Note: Now that the right hand side of the equation is 0, we can try to factor this quadratic equation.

*Since every number is divisible by 4, we can simplify the problem to:

x

^{2}+ 30x - 175 = 0*At this point you can factor or use the quadratic formula. This does factor easily:

(x - 5)(x + 35) = 0

*Split into two problems:

x - 5 = 0 OR x + 35 = 0

x = 5 OR x = -35

**Since the width of anything cannot be negative, we know the answer is x = 5. So the width of the walk is 5 feet all the way around the pool.**

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