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.
consecutive odd integers differ by 2
(n + 2)^3 - n^3 = 400 + (n+2)^2 + n^2
To solve this problem, we can first represent the two consecutive odd positive integers as x and x+2.
According to the problem, the difference of the cubes of these two integers is 400 more than the sum of their squares. So, we can set up the following equation:
(x+2)^3 - x^3 = (x+2)^2 + x^2 + 400
Now, let's simplify this equation step by step:
(x+2)^3 - x^3 = (x+2)^2 + x^2 + 400
(x^3 + 6x^2 + 12x + 8) - x^3 = x^2 + 4x + 4 + x^2 + 400
6x^2 + 12x + 8 - x^3 = 2x^2 + 4x + 404
Rearranging the terms and simplifying further:
-x^3 + 6x^2 + 12x + 8 - 2x^2 - 4x - 404 = 0
-x^3 + 4x^2 + 8x - 396 = 0
Now, we need to find the values of x that satisfy this equation. One way to do this is by using a numerical method, like guess-and-check, or by using a graphing calculator.
Using a graphing calculator, we can plot the equation y = -x^3 + 4x^2 + 8x - 396 and find the x-values where the graph intersects the x-axis, which represent the solutions.
By analyzing the graph, we can see that there is only one x-value that satisfies the equation, which is approximately x = 6.
Therefore, the two consecutive odd positive integers are 6 and 8.
To find the sum of the two integers:
6 + 8 = 14
So the sum of the two integers is 14.