Determine the sides of the base of the rectangular parallelepiped if the areas of its adjacent

Determine the sides of the base of the rectangular parallelepiped if the areas of its adjacent side faces are 15 cm2 and 20 cm2, and the diagonal of the parallelepiped is 5 √ 2.

Let the height of the rectangular parallelepiped be a, width equal to b, and length equal to c.
Then, by condition:
S1 = a * b = 15 cm ^ 2, then b = 15 / a.
S1 = a * c = 20 cm ^ 2, then c = 20 / a.
The sides of the base are equal to b and c.
By the Pythagorean theorem, the diagonal of the base is equal to:
d1 ^ 2 = b ^ 2 + c ^ 2.
The diagonal of the parallelepiped is:
d ^ 2 = a ^ 2 + d1 ^ 2 = a ^ 2 + b ^ 2 + c ^ 2.
a ^ 2 + b ^ 2 + c ^ 2 = (5√2) ^ 2 = 50.
a ^ 2 + 225 / a ^ 2 + 400 / a ^ 2 = 50.
a ^ 4 – 50a ^ 2 + 625 = 0.
We take a ^ 2 = t, then
t ^ 2 – 50t + 625 = 0.
(t – 25) ^ 2 = 0, t = 25.
Hence, a ^ 2 = t = 25, a = 5.
The sides of the base are equal
b = 15 / a = 15/5 = 3 cm,
c = 20 / a = 20/5 = 4 cm.