question bank eqns "consecutive qrs"

[guest mk]

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

[StaceyKoprince]

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.

[coltonclare]

If q, r, and s are consecutive even integers and q < r < s, which of the following CANNOT be the value of s^2 – r^2 – q^2?


Answer choices
-20
0
8
12
16

could you do the quadratic formula using -20? i.e. s^2-12s+20=...

[RonPurewal]

could you do the quadratic formula using -20? i.e. s^2-12s+20=...

Ok, I see what you're trying to do here. Presumably, you realized that q = s–4 and that r = s–2, and so now you're expanding:
s^2 – (s–2)^2 – (s–4)^2
which gives
s^2 – (s^2 – 4s + 4) – (s^2 – 8s + 16)
which is –s^2 + 12s – 20.

Note that your signs are backward, so you'd actually have to change the signs of the answer choices to get them to conform to what you wrote.

[RonPurewal]

Now if I saw this problem, on the other hand, I'd just make a list:
4^2 – 2^2 – 0^2 = 12. D is gone.
6^2 – 4^2 – 2^2 = 16. E is out.
8^2 – 6^2 – 4^2 = 12 again, but that's already gone.
10^2 – 8^2 – 6^2 = 0. Bye-bye B.
12^2 – 10^2 – 8^2 = –20. Eliminate A.

... aaaaaand we're done.

If you tried algebra and got stuck, you should have IMMEDIATELY switched to this approach. If you didn't, then your test-taking strategy needs improvement.

(If you did algebra and got it to work, then, good-- but, when you review the problem, you should still come up with this other approach, too.)

[deem432]

could you do the quadratic formula using -20? i.e. s^2-12s+20=...

Ok, I see what you're trying to do here. Presumably, you realized that q = s–4 and that r = s–2, and so now you're expanding:
s^2 – (s–2)^2 – (s–4)^2
which gives
s^2 – (s^2 – 4s + 4) – (s^2 – 8s + 16)
which is –s^2 + 12s – 20.

Note that your signs are backward, so you'd actually have to change the signs of the answer choices to get them to conform to what you wrote.

When I try to factor -s^2 +12s -20 = -20
I get: -s^2 + 12s - 20 + 20 = 0
= -s^2 + 12s = 0
And then I get stuck. Where do I go from here?

[RonPurewal]

When I try to factor -s^2 +12s -20 = -20
I get: -s^2 + 12s - 20 + 20 = 0
= -s^2 + 12s = 0
And then I get stuck. Where do I go from here?

-s(s - 12) = 0

s = 0 or s = 12

[HitenS866]

Now if I saw this problem, on the other hand, I'd just make a list:
4^2 – 2^2 – 0^2 = 12. D is gone.
6^2 – 4^2 – 2^2 = 16. E is out.
8^2 – 6^2 – 4^2 = 12 again, but that's already gone.
10^2 – 8^2 – 6^2 = 0. Bye-bye B.
12^2 – 10^2 – 8^2 = –20. Eliminate A.

... aaaaaand we're done.

If you tried algebra and got stuck, you should have IMMEDIATELY switched to this approach. If you didn't, then your test-taking strategy needs improvement.

(If you did algebra and got it to work, then, good-- but, when you review the problem, you should still come up with this other approach, too.)

Hello Ron,

Can you explain your alternate approach. I didnot understand. The Algebra method is too long to complete in 2 minutes. Why do you get C as answer since 12 is repeated ?

Thanks,
Hiten

[HitenS866]

Now if I saw this problem, on the other hand, I'd just make a list:
4^2 – 2^2 – 0^2 = 12. D is gone.
6^2 – 4^2 – 2^2 = 16. E is out.
8^2 – 6^2 – 4^2 = 12 again, but that's already gone.
10^2 – 8^2 – 6^2 = 0. Bye-bye B.
12^2 – 10^2 – 8^2 = –20. Eliminate A.

... aaaaaand we're done.

If you tried algebra and got stuck, you should have IMMEDIATELY switched to this approach. If you didn't, then your test-taking strategy needs improvement.

(If you did algebra and got it to work, then, good-- but, when you review the problem, you should still come up with this other approach, too.)

Hello Ron,

Can you explain your alternate approach. I didnot understand. The Algebra method is too long to complete in 2 minutes. Why do you get C as answer since 12 is repeated ?

Thanks,
Hiten

Hello Ron,

I get your method.

Thanks.

[RonPurewal]

ok. so it seems there are no further questions, then? (if there are, please clarify. thanks.)

[RonPurewal]

also, more importantly --
DO NOT think of "2 minutes" as a LIMIT!
2 minutes is the AVERAGE time per quant problem!
obviously, it's not reasonable to expect every data point to fall below the usual AVERAGE!
there will be some problems you should reasonably expect to finish in well below 2 minutes (especially SHORT problems that don't require any "translation" or testing of cases)... and there will also be problems that take LONGER.

...and, don't forget, good time management has nothing to do with minutes and seconds, anyway.
good time management is just
• being honest with yourself, and realizing when you're NOT MAKING PROGRESS on a problem,
and
• QUITTING what you're currently doing, AS SOON AS you get stuck.
as long as you don't sit there staring at the screen hoping for divine inspiration, and as long as you don't do "work" that isn't aimed toward any particular goal (like the way you'd do random steps in the hope of earning "partial credit" in school)... you WILL NOT experience "time pressure" on this exam. i promise you.