A square is altered so that one dimension is increased by 4 inches, while the other dimension is decreased by 2. The area of the resulting rectangle is 55 square inches. Find the area of the orginial square
(s + 4) (s - 2) = 55 ... s^2 - 2s - 63 = 0
solve for s (factoring or quadratic formula)
s^2 is the area you seek
Let's assume that the dimensions of the original square are x inches (length) and x inches (width).
According to the problem, one dimension is increased by 4 inches and the other dimension is decreased by 2 inches in the altered rectangle. So, the new dimensions of the rectangle would be (x+4) inches (length) and (x-2) inches (width).
The area of a rectangle is given by the formula: length × width. So, the area of the altered rectangle is (x+4) × (x-2) square inches.
We are given that the area of the altered rectangle is 55 square inches. So, we can set up the equation:
(x+4) × (x-2) = 55
Expanding the equation, we get:
x^2 + 2x - 8 = 55
Rearranging and simplifying the equation, we have:
x^2 + 2x - 63 = 0
To solve this quadratic equation, we can factor it or use the quadratic formula. Factoring the equation, we get:
(x + 9)(x - 7) = 0
This means that either (x + 9) = 0 or (x - 7) = 0.
Solving each equation separately, we find that x = -9 or x = 7.
Since the dimensions of a square cannot be negative, we discard the solution x = -9.
Therefore, the side length of the original square is x = 7 inches.
To find the area of the original square, we square the side length:
Area = x^2 = 7^2 = 49 square inches.
Therefore, the area of the original square is 49 square inches.
To find the area of the original square, let's set up some variables and equations.
Let's assume the original side length of the square is "x" inches. This means the original area of the square is x^2 square inches.
According to the given information, when one dimension is increased by 4 inches and the other dimension is decreased by 2 inches, we get a rectangle with an area of 55 square inches.
The new dimension of the rectangle, which was increased by 4 inches, is x + 4 inches.
The new dimension of the rectangle, which was decreased by 2 inches, is x - 2 inches.
Since the area of a rectangle is equal to the product of its dimensions, we can set up the equation:
(x + 4) * (x - 2) = 55
Simplifying this equation, we have:
x^2 + 4x - 2x - 8 = 55
Combining like terms, we get:
x^2 + 2x - 8 = 55
Rearranging the equation, we have:
x^2 + 2x - 63 = 0
To find the value of x, we can solve this quadratic equation using factoring, completing the square, or using the quadratic formula. In this case, let's use factoring.
We need to find two numbers whose product is -63 and whose sum is 2. After trying different pairs, we find that the numbers are 9 and -7.
So, we can rewrite the equation as:
(x + 9)(x - 7) = 0
Now we can set each factor equal to zero and solve for x:
x + 9 = 0 or x - 7 = 0
If we solve for x in each equation, we get:
x = -9 or x = 7
Since the side length of a square cannot be negative, we can discard the -9 solution.
Therefore, the original side length of the square is x = 7 inches.
To find the area of the original square, we square the side length:
Area = x^2 = 7^2 = 49 square inches.
So, the area of the original square is 49 square inches.