SOLUTION: 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)

Question 1201936: 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)
Found 2 solutions by ikleyn, math_tutor2020:
Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!
.
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)
~~~~~~~~~~~~~~~~~~

If + = 5, then by squaring p^2 + 2 + = 25 p^2 + = 25 - 2 = 23 p^2 - 2 + = 23 - 2 = 21 = 21 - = +/- . That is all that we can state about - . We can not state that it is necessary positive: it can be either positive or negative .

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You can check my solution and my statement directly.

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

giving the solutions    = , that are either   4.791287847 or   0.208712153.

First     value gives   p - = 4.582575695 = .

Second value gives   p - = -4.582575695 = .

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

of listed options, only (E) may happen.

But for completeness, also    - = may happen, too, in addition to options listed in the problem's list.

Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

Answer: Choice (E) sqrt(21)

Explanation:

Square both sides of the given equation
p+ (1/p) = 5
[ p+ (1/p) ]^2 = 5^2
[ p+ (1/p) ][ p+ (1/p) ] = 25
p*p + p*(1/p) + (1/p)*p + (1/p)(1/p) = 25 .... FOIL rule
p^2 + 1 + 1 + (1/p)^2 = 25
p^2 + 2 + (1/p)^2 = 25
p^2 + (1/p)^2 = 23
I'll refer to this equation as eq2 for a substitution step later on.

Let q = p - (1/p)
Square both sides to see what happens

p - (1/p) = q
[ p - (1/p) ]^2 = q^2
p*p + p*(-1/p) + (-1/p)*p + (-1/p)(-1/p) = q^2
p^2 - 1 - 1 + (1/p)^2 = q^2
p^2 - 2 + (1/p)^2 = q^2
p^2 + (1/p)^2 - 2 = q^2
[ p^2 + (1/p)^2 ] - 2 = q^2
[ p^2 + (1/p)^2 ] - 2 = q^2
[ 23 ] - 2 = q^2 ......... substitution; use eq2
21 = q^2
q^2 = 21
q = sqrt(21)
p - (1/p) = sqrt(21)

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Another approach would be to solve the equation p + (1/p) = 5 for p

p + (1/p) = 5
p * [ p + (1/p) ] = p*5
p^2 + 1 = 5p
p^2 - 5p + 1 = 0

Use the quadratic formula to find these two roots
p = (5 + sqrt(21))/2
p = (5 - sqrt(21))/2

Then use either root to compute p - (1/p)
I'll let you do these steps.

Hint:
If p = (5+sqrt(21))/2, then 1/p = (5 - sqrt(21))/2 after rationalizing the denominator.
A similar situation happens when p = (5 - sqrt(21))/2

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