Question 1201936
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If p + 1/p = 5 and p does not equal 0, which of the following is a possible value of p - 1/p ?
(A) sqrt(25)
(B) sqrt(24)
(C) sqrt(23)
(D) sqrt(22)
(E) sqrt(21)
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<pre>
If  {{{p}}} + {{{1/p}}} = 5,  then by squaring

    p^2 + 2 + {{{1/p^2}}} = 25

    p^2 + {{{1/p^2}}} = 25 - 2 = 23

    p^2 - 2 + {{{1/p^2}}} = 23 - 2 = 21

    {{{(p - 1/p)^2}}} = 21

    {{{p}}} - {{{1/p}}} = +/- {{{sqrt(21)}}}.


That is all that we can state about  {{{p}}} - {{{1/p}}}.


We can not state that it is necessary positive: it can be either positive  {{{sqrt(21)}}}  or negative  {{{-sqrt(21)}}}.
</pre>
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You can check my solution and my statement directly.


From the given equation, it implies  &nbsp;&nbsp;p^2 - 5p + 1 = 0,


giving the solutions  &nbsp;&nbsp;{{{p[1,2]}}} = {{{(5 +- sqrt(21))/2}}}, &nbsp;that are  either &nbsp;&nbsp;4.791287847 &nbsp;or  &nbsp;&nbsp;0.208712153.


First &nbsp;&nbsp;&nbsp;&nbsp;value gives  &nbsp;&nbsp;p - {{{1/p}}} = 4.582575695 = {{{sqrt(21)}}}.


Second value gives &nbsp;&nbsp;p - {{{1/p}}} = -4.582575695 = {{{-sqrt(21)}}}.



So, &nbsp;what we can state definitely, &nbsp;is that &nbsp;(A), &nbsp;(B), &nbsp;(C) &nbsp;and &nbsp;(D) &nbsp;never may happen;


of listed options, &nbsp;only &nbsp;(E) &nbsp;may happen.


But for completeness, &nbsp;also  &nbsp;&nbsp;{{{p}}} - {{{1/p}}} = {{{-sqrt(21)}}} &nbsp;may happen, &nbsp;too, &nbsp;in addition to options listed in the problem's list.