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# If p is a negative number and 0 < s < |p|, which of the following

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Math Expert
Joined: 02 Sep 2009
Posts: 61534
If p is a negative number and 0 < s < |p|, which of the following  [#permalink]

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24 May 2018, 04:12
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If p is a negative number and $$0 < s < |p|$$, which of the following must also be a negative number?

A) $$(p+s)^2$$

B) $$(p-s)^2$$

C) $$(s-p)^2$$

D) $$p^2-s^2$$

E) $$s^2-p^2$$

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Re: If p is a negative number and 0 < s < |p|, which of the following  [#permalink]

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24 May 2018, 04:22
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Bunuel wrote:
If p is a negative number and $$0 < s < |p|$$, which of the following must also be a negative number?

A) $$(p+s)^2$$

B) $$(p-s)^2$$

C) $$(s-p)^2$$

D) $$p^2-s^2$$

E) $$s^2-p^2$$

As all we're given are variables, we'll pick simple numbers to eliminate incorrect answers.
This is an Alternative approach.

Then
(A) is (-1)^2 = 1. NO
(B) is (-3)^2 = 9. NO
(C) is 3^2 = 9. NO
(D) is (-2)^2 - 1^2 = 4 - 1 = 3. NO
(E) must be our answer. Let's verify: 1^2 - (-2)^2 = 1 - 4 = -3.

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Re: If p is a negative number and 0 < s < |p|, which of the following  [#permalink]

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24 May 2018, 21:10
E;

In the number line value of S will be between 0 and +Mod p,

Therefore P2-s2 will always be negative.
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Re: If p is a negative number and 0 < s < |p|, which of the following  [#permalink]

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12 Aug 2018, 06:55
Faster way:
First three answers are wrong. Elevating to the two the whole sum will lead for sure to a positive number.
For the least two answers (D and E) we know that both of the numbers ( p and s ) will be positive due to the elevation to the power of two. We know as well that p is greater than s so if we want a negative sum we need to subtract p to s.
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Re: If p is a negative number and 0 < s < |p|, which of the following  [#permalink]

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18 Jan 2020, 02:27
Based on the question we know that S is positive, P is negative, but the absolute value of P is greater than S. This means no matter what S is, P squared will be greater than S squared.

The first three answers we can disregard immediately because they are all a sum or difference of S & P but it is squared. So no matter what actual sum or difference is the result squared will be positive.

Now we are left with only D & E as possible solutions to the problem. D is P^2 - S^2, and because the absolute value of P is greater than S, we know that P^2 will be greater than S^2. Example, if S =2, then P is a negative number that is less than -2. So let's assume it's -3.

-3^2 - 2^2 = 9 - 4 = 5. It's positive, and no matter what we substitute in for P & S, as long as the absolute value of P is greater than S, and as such P squared will always be greater than S squared.

The only option left is E.
Re: If p is a negative number and 0 < s < |p|, which of the following   [#permalink] 18 Jan 2020, 02:27
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