A Quadratic Word Problem
If one side of a square is increased by 2 inches and an adjacent side is decreased by 2 inches, the area of the resulting rectangle is 32 square inches. Find the length of one side of the square.
a = length of one side of the square
W = Rectangle width
H = Rectangle height
A = Rectangle area
W = a + 2
H = a - 2
A = W * H
A = ( a + 2 ) * ( a - 2 ) = 32
32 = a ^ 2 + 2 a - 2 a - 4
32 = a ^ - 4 Add 4 to both sides
32 + 4 = a ^ - 4 + 4
36 = a ^ 2
a = sqrt ( 36 ) = 6
The length of one side of the square = 6 in
6 inch
To solve this quadratic word problem, we need to follow a few steps:
Step 1: Let's assume the length of one side of the square is x inches.
Step 2: If one side of the square is increased by 2 inches, then the length of the rectangle will be (x + 2) inches.
Step 3: Similarly, if an adjacent side is decreased by 2 inches, then the width of the rectangle will be (x - 2) inches.
Step 4: The area of a rectangle is given by the formula: Area = Length × Width. So, the area of the rectangle is (x + 2) × (x - 2).
Step 5: According to the problem, the area of the rectangle is 32 square inches. So, we can set up the equation: (x + 2) × (x - 2) = 32.
Now, let's solve this equation to find the length of one side of the square:
(x + 2) × (x - 2) = 32
(x^2 - 2x + 2x - 4) = 32
(x^2 - 4) = 32
x^2 = 32 + 4
x^2 = 36
Taking the square root of both sides, we get:
x = ±√36
x = ±6
Since the side length of a square cannot be negative, we discard the negative value.
Therefore, the length of one side of the square is 6 inches.