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.
Let's assume that the two consecutive odd positive integers are x and x+2.
According to the problem, the difference of their cubes is 400 more than the sum of their squares. So we can write the equation:
(x+2)^3 - x^3 = 400 + x^2 + (x+2)^2
Expanding the cubes and simplifying, we get:
(x^3 + 6x^2 + 12x + 8) - x^3 = 400 + x^2 + (x^2 + 4x + 4)
Simplifying further, we have:
6x^2 + 12x + 8 - x^3 = 400 + 2x^2 + 4x + 4
Rearranging the terms, we get:
x^3 - 4x^2 - 8x - 396 = 0
Now, we can find the roots of this equation using a calculator or by factoring. By factoring, we can see that one of the roots is x = 9.
So the consecutive odd positive integers are 9 and 9+2 = 11.
The sum of the two integers is 9 + 11 = 20. Therefore, the sum of the two integers is 20.
To solve this problem, let's start by assigning variables to the unknowns.
Let's say the two consecutive odd positive integers are represented by n and n+2.
According to the given information, the difference of their cubes is 400 more than the sum of their squares. Mathematically, we can represent this information as:
(n+2)^3 - n^3 = (n^2 + (n+2)^2) + 400
Now, let's simplify this equation step by step.
Expanding (n+2)^3:
(n^3 + 3n^2 + 6n + 8) - n^3 = (n^2 + (n^2 + 4n + 4)) + 400
Cancelling out similar terms:
(n^2 + 6n + 8) = (2n^2 + 4n + 4) + 400
Expanding (2n^2 + 4n + 4):
n^2 + 6n + 8 = 2n^2 + 4n + 404
Rearranging the equation:
2n^2 - n^2 + 4n - 6n + 8 - 404 = 0
Combining like terms:
n^2 - 2n - 396 = 0
Now we have a quadratic equation. We can solve this equation by factoring, completing the square, or using the quadratic formula. Let's use factoring in this case.
Factoring the quadratic equation:
(n - 18)(n + 22) = 0
Setting each factor equal to zero and solving for n:
n - 18 = 0 or n + 22 = 0
n = 18 or n = -22
Since we are looking for positive odd integers, we discard the negative value. Therefore, n = 18.
Now that we have the value of n, we can find the consecutive odd positive integers:
n = 18
n+2 = 18 + 2 = 20
So, the two consecutive odd positive integers are 18 and 20.
To find the sum of these two integers:
18 + 20 = 38
Therefore, the sum of the two integers is 38.
Let the words numbers be be (x+2) and (x)
then we get equation
(x+2)^3 - x^3 - (2x^2 + 4x+ 4) = 400
(not solving further as this a sum of brilliant's this week challenge0