a^3 - b^3 = 400+a^2+b^2
also, a=b+2, so
(b+2)^3 - b^3 = (b+2)^2 + b^2 + 400
b=9, so a=11
11^3-9^3 = 1331-729 = 602
11^2+9^2+400 = 602
also, a=b+2, so
(b+2)^3 - b^3 = (b+2)^2 + b^2 + 400
b=9, so a=11
11^3-9^3 = 1331-729 = 602
11^2+9^2+400 = 602
According to the problem, the difference of the cubes of the two integers is 400 more than the sum of their squares. We can write this as an equation:
(x + 2)^3 - x^3 = x^2 + (x + 2)^2 + 400
Now, let's simplify the equation:
(x^3 + 6x^2 + 12x + 8) - x^3 = x^2 + (x^2 + 4x + 4) + 400
Simplifying further:
6x^2 + 12x + 8 - x^3 = 2x^2 + 4x + 4 + 400
Combining like terms:
5x^3 - 4x^2 - 8x - 396 = 0
At this point, we have a cubic equation. To find the sum of the two integers, we need to find the values of x and x + 2 that satisfy this equation.
Solving a cubic equation can be a bit complicated, but we can use numerical methods or approximations to find an approximate solution. One popular method is using a graphing calculator or an algebraic software program.
Alternatively, we can try to use trial and error to find integer solutions. We can start by testing some small values of x that are odd, such as x = 1, 3, 5, etc. Then, substitute these values back into the equation, and see if any of them yield a valid solution.
For example, let's try x = 1:
5(1)^3 - 4(1)^2 - 8(1) - 396 = 0
5 - 4 - 8 - 396 = 0
-403 ≠0
Since x = 1 does not yield a valid solution, let's try the next value, x = 3:
5(3)^3 - 4(3)^2 - 8(3) - 396 = 0
405 - 36 - 24 - 396 = 0
-51 ≠0
Again, x = 3 does not yield a valid solution. We can continue this process until we find a valid solution.
Please note that this approach may or may not lead to an exact solution, and it can be time-consuming. Numerical methods or approximations are typically used in situations where an exact solution is not easily obtainable.