SOLUTION: A rectangular piece of​ cardboard, whose area is 396 square​ centimeters, is made into an open box by cutting a 2​-centimeter square from each corner and turning up the sides

Let x and y be the values under the question, i.e. dimensions of the original piece of cardboard.


The height of the box is 2 cm;

the area of the base of the box is equal to its volume divided by the height

    area of the base of the box =  = 252 cm^2.


It gives you your first equation


    (x-2*2)*(y-2*2) = 252,   or

    (x-4)*(y-4) = 252.       (1)


The second equation is for the surface area of the box

    (x-4)*(y-4) + 2*(x-4) + 2*(y-4) = 396 - 4*(2*2)  cm^2,   or

    (x-4)*(y-4) + 2*(x-4) + 2*(y-4) = 380.    (2)


It is your second equation.


Simplify them.


In the left side of (2),  replace  (x-4)*(y-4) by 252, based on (1).

You will get then instead of (2)

    252 + 2*(x-4) + 2*(y-4) = 380,   or

    2*(x-4) + 2*(y-4) = 380 - 252 = 128,

    (x-4) + (y-4) = 64.


So, the sum  (x-4) + (y-4) = 64,  while the product  (x-4)*(y-4) = 252.


Thus (x-4) and (y-4) are the roots of the quadratic equation  t^2 - 64t + 252 = 0.


Hence,   =  =  =  = .


It gives  x-4 = ,  x =  = 63.785 cm.

          y-4 = ,  y =  =  8.215 cm.


CHECK.  Equation (1) : (x-4)*(y-4) = (63.785 -4)*(8.215-4) = 251.994 = 252 cm^2;

        Equation (2) : (x-4)*(y-4) + 2*(x-4) + 2*(y-4) = (63.785 -4)*(8.215-4) + 2*(63.785 -4) + 2*(8.215-4) = 379.994 = 380 cm^2.


ANSWER.  The dimensions are  x =  = 63.785 cm  and  y =  =  8.215 cm.