the area of a square is numerically 165 more than the perimeter. find the length of the side
If x = side, then
x^2 = area
4x = perimeter
Therefore,
x^2 - 4x = 165
x^2 - 4x - 165 = 0
Solve for x.
To solve this problem, we can start by writing down what we know.
Let's assume that the side length of the square is 's'.
Given that the area of the square is numerically 165 more than the perimeter, we can write the equation:
Area of square = Perimeter of square + 165
The area of the square is equal to the side length squared (s^2) and the perimeter is equal to 4 times the side length (4s). So, we can rewrite the equation as:
s^2 = 4s + 165
Now, let's solve this quadratic equation to find the length of the side (s).
Rearrange the equation:
s^2 - 4s - 165 = 0
To solve this quadratic equation, we can either factorize it or use the quadratic formula.
Since the equation cannot be easily factorized, let's use the quadratic formula:
s = (-b ± √(b^2 - 4ac)) / (2a)
In this equation, the coefficient 'a' is the coefficient of s^2 term, which is 1 in our case. The coefficient 'b' is the coefficient of the s term, which is -4, and the constant term 'c' is -165.
Plugging these values into the quadratic formula, we have:
s = (-(-4) ± √((-4)^2 - 4(1)(-165))) / (2(1))
Simplifying,
s = (4 ± √(16 + 660)) / 2
s = (4 ± √676) / 2
s = (4 ± 26) / 2
Now we have two potential solutions for 's':
s1 = (4 + 26) / 2 = 30 / 2 = 15
s2 = (4 - 26) / 2 = -22 / 2 = -11
Since the side length of a square cannot be negative, we discard s2 = -11 and conclude that the length of the side, s, is 15 units.