a square with perimeter 5 less then the area. perimeter and area equal 45
To solve this problem, we need to set up equations based on the given information.
Let's assume that the side length of the square is 's'. The perimeter of a square is found by multiplying the side length by 4, so the perimeter of the square can be expressed as 4s.
According to the given information, the perimeter of the square is 5 less than its area. Mathematically, this is written as:
4s = s^2 - 5
We also know that the sum of the perimeter and the area is 45. So, we can set up another equation:
4s + s^2 = 45
Now, we have a system of equations:
4s = s^2 - 5
4s + s^2 = 45
To find the side length of the square, we can solve this system of equations.
First, let's rearrange the equation 4s = s^2 - 5 to have everything on one side:
s^2 - 4s - 5 = 0
Next, we solve this quadratic equation by factoring, completing the square, or using the quadratic formula. Let’s use the quadratic formula in this case:
s = (-b ± √(b^2 - 4ac)) / 2a
For our equation, a = 1, b = -4, and c = -5, so we can substitute these values into the quadratic formula:
s = (-(-4) ± √((-4)^2 - 4(1)(-5))) / 2(1)
s = (4 ± √(16 + 20)) / 2
s = (4 ± √36) / 2
s = (4 ± 6) / 2
Now, we have two possibilities for the side length:
s₁ = (4 + 6) / 2 = 10 / 2 = 5
s₂ = (4 - 6) / 2 = -2 / 2 = -1
Since a side length cannot be negative, we can discard the second value, s₂ = -1.
Therefore, the side length of the square is s = 5.