Final answer:
To find the value of k, we use the fact that the sum of deviations from the mean is zero. After setting up the equation with the given deviations, combining like terms, and then factoring the resulting quadratic equation, we determine that k = -4/3.
Step-by-step explanation:
The question asks us to find the value of k given that the deviations from the mean of a set of numbers are (k+3)2, (k+3)2, -2, k, and (k+2)2. Since the sum of the deviations from the mean is always zero, we can set up the following equation:
(k+3)2 + (k+3)2 + (-2) + k + (k+2)2 = 0
Combining like terms and simplifying the squares, we get:
2(k+3)2 + (k+2)2 + k - 2 = 0
Expanding the squares, we have:
2(k2 + 6k + 9) + (k2 + 4k + 4) + k - 2 = 0
2k2 + 12k + 18 + k2 + 4k + 4 + k - 2 = 0
Combining like terms yields:
3k2 + 17k + 20 = 0
We can solve this quadratic equation by factoring or using the quadratic formula. Upon factoring, we find:
(3k + 4)(k + 5) = 0
Therefore, k could be -4/3 or -5, but since the deviations are squared and we usually take the principal square root, we disregard the negative values, which would leave us with the following value for k:
k = -4/3
In this case, -4/3 satisfies the deviation equations, so this is our answer for the value of k.