The area of a square is 45 more than the perimeter. How do you find the length of the side?

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Answer 1 · 2016-01-15T13:44:31

Length of one side is 9 units.
Rather than doing a straight factorising approach I have used the formula to demonstrate its use.

As it is a square the length of all the sides is the same.
Let the length of 1 side be L
Let the area be A

Then #A=L^2#............................(1)

Perimeter is #4L#........................(2)

The question states: "The area of a square is 45 more than.."

#=> A=4L+45#.................................(3)

Substitute equation (3) into equation (1) giving:

#A=4L+45=L^2....................(1_a)#

So now we are able to write just 1 equation with 1 unknown, which is solvable.
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

#4L+45=L^2#

Subtract #L^2# from both sides giving a quadratic.

#-L^2+4L+45=0#
The conditions that satisfy this equation equalling zero gives us the potential size of L

Tony B

Using #ax+bx+c=0# where #x= (-b+-sqrt(b^2-4ac))/(2a)#

#a=-1#
#b=4#
#c=45#

#x=(-4+-sqrt((4)^2-4(-1)(45)))/(2(-1))#

#x=(-4+-14)/(-2)#

#x= (-18)/(-2) = +9#

#x=(+10)/(-2)=-5#

Of these two #x=-5# is not a logical length of side so

#x=L=9#

#"Check "-> A= 9^2= 81 "units"^2#

#4L= 36 -> 81-36= 45#

So area does indeed equal sum of sides + 45