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Step-by-step explanation:
the original side length is x.
the volume of a block is length × width × height.
so, for a cube with all sides being equal, the volume in our case is x³ (x × x × x).
now, with the same formula but with every side shortened by 1 cm we get
(x-1)(x-1)(x-1) = x³ - 91 (the original volume minus 91)
now, we have to do the multiplications.
(x-1)(x-1) = x² - 2x + 1
(x²-2x+1)(x-1) = x³ - x² - 2x² + 2x + x - 1 = x³ -3x² + 3x -1
x³ - 3x² + 3x - 1 = x³ - 91
-3x² + 3x = -90
-x² + x = -30
x² - x = 30
x² - x - 30 = 0
the general solution to such a squared equation is
x = (-b ± sqrt(b² - 4ac))/(2a)
in our case
a = 1
b = -1
c = -30
x = (1 ± sqrt((-1)² - 4×1×-30))/(2×1) =
= (1 ± sqrt(1 + 120))/2 = (1 ± sqrt(121))/2 =
= (1 ± 11)/2
x1 = (1 + 11)/2 = 12/2 = 6
x2 = (1 - 11)/2 = -10/2 = -5
a negative number does not make any sense for a side length, so our solution is x = 6 cm.
the original side length of the cube is 6 cm.
to check :
6³ = 216
(6-1)³ = 5³ = 125
216 - 125 = 91
perfect.