Finding the range of a set using quadratics? - MPOTW

Arithmetic Mean

Given that the mean of this set of numbers is 10, what are the possible ranges?

{2 x - 7, x, {(x - 1)}^{2}, x^{2}, 5 x}

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Using the definition of mean and the set of numbers given we get:

\frac{(2 x - 7 + x + {(x - 1)}^{2} + x^{2} + 5 x)}{5} = 10

Simplifying ...

2 x - 7 + x + {(x - 1)}^{2} + x^{2} + 5 x = 50

8 x - 7 + (x^{2} - 2 x + 1) + x^{2} = 50

2 x^{2} - 6 x - 56 = 0

This is a quadratics. Simplifying and solving ...

x = {-4, 7}

So, this gives us two posibilities for the range.

For $x = -4$ gives this set of values ${-15, -4, 25, 16, -20}$ resulting in a range of $25 - -20 = 45$

For $x = 7$ gives this set of values ${7, 7, 36, 49, 35}$ resulting in a range of $49 - 7 = 42$

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