The difference of the cubes of two consecutive odd positive integers is 400 more than the sum of their squares. Find the sum of the two integers.
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
To solve this problem, let's first express the odd positive integers in terms of variables. Let's assume the smaller integer is x, and the larger integer is x + 2 (since they are consecutive odd integers).
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.