I have two questions in bold, please help.


If q, r, and s are consecutive even integers and q < r < s, which of the following CANNOT be the value of s2 – r2 – q2?

Ans: If q, r, and s are consecutive even integers and q < r < s, then r = s – 2 and q = s – 4. The expression s2– r2– q2 can be written as s2– (s –2)2 – (s – 4)2.

If we multiply this out, we get:

s^2– (s –2)^2 – (s – 4)^2 =
s^2– (s^2 – 4s + 4) – (s^2 – 8s + 16) =
s^2– s^2 + 4s – 4 – s^2 + 8s – 16 =
-s^2 + 12s – 20

The question asks which of the choices CANNOT be the value of the expression -s2 + 12s – 20 so we can test each answer choice to see which one violates what we know to be true about s, namely that s is an even integer.

Then it goes on to test them. But I think the factoring is wrong

Testing (E) we get:
-s2 + 12s – 20 =16
-s2 + 12s – 36 = 0
s2+ 12s – 36 = 0
(s – 6)(s – 6) = 0 But, (s-6) (s-6) = s^2-12s+36 Are the signs messed up or did they multiply by neg 1 and then forget to change them? The same is true of the other tests.
s = 6. This is an even integer so this works.

Testing (D) we get:
-s2 + 12s – 20 =12
-s2 + 12s – 32 = 0
s2+ 12s – 32 = 0
(s – 4)(s – 8) = 0
s = 4 or 8. These are even integers so this works.

Testing (C) we get:
-s2 + 12s – 20 = 8
-s2 + 12s – 28 = 0
s2+ 12s – 28 = 0
Since there are no integer solutions to this quadratic (meaning there are no solutions where s is an integer), 8 is not a possible value for the expression.

The second question I have is: For q< r < s can't you do it this way:

q= s

r= s+2

s= s+4

s^2-r^2-q^2

(s+4)^2-(s+2)^2-s^2 = -s^2+4s +12

-1 (-s^2+4s+12) = s^2 -4s -12

Thank you

Please remember to post not only the question but also the answer choices - I need to see the full text to determine the best way to approach the problem.

Also, please check your text to make sure that you use the carat symbol (^) any time you need to indicate an exponent. It seems as though this might not have been done consistently.