Find three consecutive integers odd integers such that the square of the first increased by twice the second is equal to two less than three times the third
If the smallest is x, then
x^2 + 2(x+2) = 3(x+4)-2
Unfortunately, this will not work (why?)
To make sure x is odd, we need x = 2k+1
Then we solve
(2k+1)^2 + 2(2k+3) = 3(2k+5)-2
To solve this problem, let's assume the first odd integer as 'x'. Since we are looking for three consecutive odd integers, the second odd integer will be 'x + 2', and the third odd integer will be 'x + 4'.
According to the problem statement, the square of the first integer increased by twice the second integer is equal to two less than three times the third integer. We can represent this in the form of an equation:
x^2 + 2(x + 2) = 3(x + 4) - 2
Let's solve this equation step by step:
Expand the terms:
x^2 + 2x + 4 = 3x + 12 - 2
Combine like terms:
x^2 + 2x + 4 = 3x + 10
Move all the terms to one side to obtain a quadratic equation:
x^2 - x - 6 = 0
We can factor this quadratic equation:
(x - 3)(x + 2) = 0
Setting each factor to zero, we find two possible values for 'x':
x - 3 = 0 --> x = 3
x + 2 = 0 --> x = -2
Since we are looking for odd consecutive integers, we can discard the value 'x = -2' as it is not odd.
Therefore, the first odd integer is x = 3, the second odd integer is 3 + 2 = 5, and the third odd integer is 3 + 4 = 7.
So, the three consecutive odd integers that satisfy the given conditions are 3, 5, and 7.
Let's represent the three consecutive odd integers as n, n + 2, and n + 4.
According to the problem, we have:
(n)^2 + 2(n + 2) = 3(n + 4) - 2.
Expanding the equation:
n^2 + 2n + 4 = 3n + 12 - 2.
Simplifying further:
n^2 + 2n + 4 = 3n + 10.
Moving all terms to one side:
n^2 - n - 6 = 0.
Factoring the quadratic equation:
(n - 3)(n + 2) = 0.
Setting each factor equal to zero:
n - 3 = 0 or n + 2 = 0.
For n - 3 = 0, we get n = 3.
For n + 2 = 0, we get n = -2.
Since we need three consecutive odd integers, we will discard n = -2, as it is not odd.
Therefore, the three consecutive odd integers are:
n = 3,
n + 2 = 3 + 2 = 5,
n + 4 = 3 + 4 = 7.
So the three consecutive odd integers are 3, 5, and 7.