Find 3 consecutive positive integers whose product is 5 more than the cube of the middle number.
Tried x*(x+1)*(x+2) = (x+1)^3 + 5, but that solves to x = -6 which isn't positive.
4*5*6 = 120 = 5^3-5
-6 * -5 * -4 = -120 = -5^3 + 5
I suspect a typo in the problem.
To find the consecutive positive integers, we can try a different approach. Let's assume the middle number is "x."
According to the given information, the product of the three consecutive positive integers is 5 more than the cube of the middle number. So, we have the equation:
x * (x + 1) * (x + 2) = x^3 + 5
Expanding the left side of the equation:
x^3 + 3x^2 + 2x = x^3 + 5
Subtracting x^3 from both sides of the equation:
3x^2 + 2x = 5
Rearranging the equation:
3x^2 + 2x - 5 = 0
Now we can solve this quadratic equation using factoring or the quadratic formula. However, in this case, the equation does not factor nicely. Therefore, we will use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Here, a = 3, b = 2, and c = -5. Plugging these values into the formula, we get:
x = (-2 ± √(2^2 - 4 * 3 * -5)) / (2 * 3)
Simplifying further:
x = (-2 ± √(4 + 60)) / 6
x = (-2 ± √64) / 6
x = (-2 ± 8) / 6
Now we have two possible solutions:
x = (8 - 2) / 6 = 6 / 6 = 1
x = (-8 - 2) / 6 = -10 / 6 = -5/3
Since we are looking for positive integers, the value of x = -5/3 is not valid. Therefore, the middle number is x = 1.
The three consecutive positive integers are 1, 2, and 3.
To find the three consecutive positive integers, we can approach the problem step by step.
Let's assume the middle number is represented by x. Then the three consecutive positive integers can be written as x-1, x, and x+1.
According to the given information, the product of these three numbers should be 5 more than the cube of the middle number. We can represent this as an equation:
(x - 1) * x * (x + 1) = x^3 + 5
Now we can solve this equation to find the value of x that satisfies the condition.
Expanding and simplifying the left side of the equation:
x^3 - x = x^3 + 5
Moving all terms to one side of the equation:
0 = x^3 - x^3 + 5 + x
0 = 5 + x
Subtracting x from both sides:
-x = 5
Multiplying both sides by -1:
x = -5
At this point, we have found that the middle number is -5, which is not a positive integer as required by the question.
Therefore, based on the given condition, there are no three consecutive positive integers whose product is 5 more than the cube of the middle number.