A square and rectangle have the same area. The length of the rectangle is five inches more than twice the length of the side of the square. The width of the rectangle is 6 inches less than the side of the square. Find the length of the side of the square.
Let's assume that the side length of the square is "s" inches.
According to the given information, the area of the square is equal to the area of the rectangle, so we can set up the following equation:
Area of square = Area of rectangle
s^2 = (2s+5)(s-6)
Now, let's solve this equation step-by-step:
Expand the right side of the equation:
s^2 = 2s^2 + 5s - 12s - 30
Combine like terms:
s^2 = 2s^2 - 7s - 30
Rearrange the equation:
0 = 2s^2 - s^2 - 7s - 30
Combine like terms:
0 = s^2 - 7s - 30
Now, we have a quadratic equation. Let's solve it by factoring or using the quadratic formula:
(s - 10)(s + 3) = 0
Setting each factor equal to zero:
s - 10 = 0 or s + 3 = 0
s = 10 or s = -3
Since the side length of a geometric figure cannot be negative, we can discard the solution s = -3.
Therefore, the side length of the square is 10 inches.
To find the length of the side of the square, we can set up a system of equations using the given information.
Let's denote the length of the side of the square as "s". Therefore, the area of the square is s^2.
According to the problem, the length of the rectangle is five inches more than twice the length of the side of the square. This can be written as:
Length of the rectangle = 2s + 5
The width of the rectangle is 6 inches less than the side of the square. This can be expressed as:
Width of the rectangle = s - 6
Since the area of a rectangle is given by multiplying the length and the width, we can set up an equation based on the areas of the square and the rectangle, which are equal:
s^2 = (2s + 5) * (s - 6)
Now, we can solve this equation to find the length of the side of the square.
Expanding the right side of the equation:
s^2 = (2s^2 - 7s - 30)
Rearranging the equation:
0 = 2s^2 - 7s - 30 - s^2
Combining like terms:
0 = s^2 - 7s - 30
Now, we have a quadratic equation. We can solve it by factoring or using the quadratic formula.
Factoring the equation:
0 = (s - 10)(s + 3)
Setting each factor equal to zero:
s - 10 = 0 or s + 3 = 0
Solving for "s":
s = 10 or s = -3
Since the length of a side cannot be negative, we discard the solution s = -3.
Therefore, the length of the side of the square is s = 10 inches.
(2s + 5)(s - 6) = s^2
2s^2 - 7s - 30 = s^2
s^2 - 7s - 30 = 0
(s - 10)(s + 3) = 0