Finding a length

Question

The area of a square is 45 more than its perimeter.find the length of a side

Answer 1

Area = 45+perimeter

Area = side squared

A=s²

Perimeter = 4 sides = 4s

s²=45+4s

rearrange terms to get s²-4s-45=0

solve for s in one of three ways: factor, complete the square or use the quadratic formula

45 factors into 9 x 5. factors are (s-9) and (s+5)

set each factor equal to zero s-9=0 and s+5=0

s=9 or -5. sides are positive, so the answer is the sides are 9

That seems to be the intended answer: 9

But the area of a square in this problem is 81, 9 squared, 81 square feet or whatever the units are. The area is in two dimensions.

The perimeter is 4x9=36 feet or 36 one dimensional units.

You can't really compare the two. As the Big Bang's Professor Sheldon Cooper once told Penny, Light Years is distance, not time. Foot pounds is force, not weight. And he might have added, two dimensional units are not one dimensional units.

While negative lengths might preclude -5 as a side, historically, mathematicians once rejected all negative numbers. But we began accepting them. There might be some other universe with negative lengths with a real application. Just as imaginary numbers are used in electricity, there may be some practical use for negative lengths which seem more acceptable than imaginary numbers.

Answer 2

Since a square has all sides that are equal, the perimeter is defined to be x+x+x+x = 4x, where x is the length of the side of the square. Likewise, the area is x^2. So, if we take these two pieces of information together, along with the fact that the area is 45 more than the perimeter we can setup the equation x^2 = 4x+45

Subtracting 4x and 45 to the left side, we end up with the equation x^2-4x-45= 0 which is a quadratic that can factor.

When looking for the factors we want two numbers that multiply to be -45 and add up to be -4. These two numbers are -9 and 5.

Thus, we can factor our equation into (x-9)(x+5)=0. Setting each of these factors to zero (via the zero product property) we arrive with answers of x = 9 or x =-5. Clearly, x=-5 doesn't make sense in terms of a side length, so it is extraneous.

Therefore, the length of the side of the square is 9.

Answer 3

The formula for area of a square is A=s².

The formula for perimeter for any shape is to add up all of the sides. For a square, we can write that as P = 4s.

Let A = area

45 more than means 45 +

Let P = perimeter

So the equation will be:

A = 45 + P

Substitute in the formulas for area and perimeter of a square:

s² = 45 + 4s

Move everything over to one side so it is equal to 0:

s² - 4s - 45 = 0

Factor:

(s - 9)(s + 5) = 0

Solve:

s = 9 and s = -5

A side length can’t be negative, so reject s = -5.

so the side length is 9.

Lets check our answer:

Area = s²

9² = 81

Perimeter = 4s

4(9) = 36

36 + 45 does equal 81.