Question

Find two numbers such that their difference and also the difference of their cubes are given numbers; say, their difference is 6 and the difference of their cubes is 504. [Hint: Call the numbers x + 3 and x − 3.]

Answer 1

Explanation:

We are to find two numbers such that their difference and also the difference of their cubes are given numbers

Let the two numbers be x+3 and x-3

so that the difference between the numbers is 6.

Difference between the cubes

=
`(x+3)^3-(x-3)^3\\= (x+3-x+3)[(x+3)^2+(x-3)^2-(x+3)(x-3)] = 504\\3x^2+9=84\\x^2 = 25\\x = 5 or -5`

So the numbers are either 2 and 8 or

(-2 and -8)