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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?
- a)-20
- b)0
- c)8
- d)12
- e)16
Correct answer is option 'C'. Can you explain this answer?
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If q, r, and s are consecutive even integers and q < r < s, whic...
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:
s2– (s –2)2 – (s – 4)2 = s2– (s2 – 4s + 4) – (s2 – 8s + 16) = s2– s2 + 4s – 4 – s2 + 8s – 16 =
-s2 + 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.
Testing (E) we get:
-s2 + 12s – 20 =16
-s2 + 12s – 36 = 0
s2+ 12s – 36 = 0
(s – 6)(s – 6) = 0
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.
s2– (s –2)2 – (s – 4)2 = s2– (s2 – 4s + 4) – (s2 – 8s + 16) = s2– s2 + 4s – 4 – s2 + 8s – 16 =
-s2 + 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.
Testing (E) we get:
-s2 + 12s – 20 =16
-s2 + 12s – 36 = 0
s2+ 12s – 36 = 0
(s – 6)(s – 6) = 0
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.
-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.
Alternately, we could choose values for q, r, and s and then look for a pattern with our results. Since the answer choices are all within twenty units of zero, choosing integer values close to zero is logical. For example, if q = 0, r = 2, and s = 4, we get 42 – 22 – 02 which equals
16 – 4 – 0 = 12. Eliminate answer choice D.
16 – 4 – 0 = 12. Eliminate answer choice D.
Since there is only one value greater than 12 in our answer choices, it makes sense to next test q = 2, r = 4, s = 6. With these values, we get 62 – 42 – 22 which equals 36 – 16 – 4 = 16. Eliminate answer choice E.
We have now eliminated the two greatest answer choices, so we must test smaller values for q, r, and s. If q = -2, r = 0, and s = 2, we get 22 – 02 – 22 which equals 4 – 0 – 4 = 0. Eliminate answer choice B.
At this point, you might notice that as you choose smaller (more negative) values for q, r, and s, the value of s2 < r2 < q2. Thus, any additional answers will yield a negative value. If not, simply choose the next logical values for q, r, and s: q = -4, r = -2, and s = 0. With these values we get 02 – (-2)2 – (-4)2 = 0 – 4 – 16 = -20. Eliminate answer choice A.
The correct answer is C.
Most Upvoted Answer
If q, r, and s are consecutive even integers and q < r < s, whic...
Understanding Consecutive Even Integers
Let’s define the consecutive even integers:
- q = n (the first even integer)
- r = n + 2 (the second even integer)
- s = n + 4 (the third even integer)
Calculating s² - r² - q²
We can express s² - r² - q²:
- s² = (n + 4)² = n² + 8n + 16
- r² = (n + 2)² = n² + 4n + 4
- q² = n²
Now, substituting these into the expression:
s² - r² - q² = (n² + 8n + 16) - (n² + 4n + 4) - n²
This simplifies to:
- s² - r² - q² = n² + 8n + 16 - n² - 4n - 4 - n²
- = 4n + 12
Possible Values
The expression 4n + 12 can take various values based on the integer n. It is essential to note that:
- Since n is an even integer, 4n is also even.
- Thus, 4n + 12 is even.
Evaluating the Options
Now we evaluate the options provided:
- a) -20 (even)
- b) 0 (even)
- c) 8 (even)
- d) 12 (even)
- e) 16 (even)
Among these, all are even numbers. However, we need to check which cannot be expressed as 4n + 12.
Finding n for each option
1. For -20: 4n + 12 = -20
- 4n = -32 => n = -8 (valid)
2. For 0: 4n + 12 = 0
- 4n = -12 => n = -3 (not valid, as n must be even)
3. For 8: 4n + 12 = 8
- 4n = -4 => n = -1 (not valid)
4. For 12: 4n + 12 = 12
- 4n = 0 => n = 0 (valid)
5. For 16: 4n + 12 = 16
- 4n = 4 => n = 1 (not valid)
Conclusion
Since n must always be an even integer, the value that cannot be achieved is option C) 8, as it yields an invalid n.
Let’s define the consecutive even integers:
- q = n (the first even integer)
- r = n + 2 (the second even integer)
- s = n + 4 (the third even integer)
Calculating s² - r² - q²
We can express s² - r² - q²:
- s² = (n + 4)² = n² + 8n + 16
- r² = (n + 2)² = n² + 4n + 4
- q² = n²
Now, substituting these into the expression:
s² - r² - q² = (n² + 8n + 16) - (n² + 4n + 4) - n²
This simplifies to:
- s² - r² - q² = n² + 8n + 16 - n² - 4n - 4 - n²
- = 4n + 12
Possible Values
The expression 4n + 12 can take various values based on the integer n. It is essential to note that:
- Since n is an even integer, 4n is also even.
- Thus, 4n + 12 is even.
Evaluating the Options
Now we evaluate the options provided:
- a) -20 (even)
- b) 0 (even)
- c) 8 (even)
- d) 12 (even)
- e) 16 (even)
Among these, all are even numbers. However, we need to check which cannot be expressed as 4n + 12.
Finding n for each option
1. For -20: 4n + 12 = -20
- 4n = -32 => n = -8 (valid)
2. For 0: 4n + 12 = 0
- 4n = -12 => n = -3 (not valid, as n must be even)
3. For 8: 4n + 12 = 8
- 4n = -4 => n = -1 (not valid)
4. For 12: 4n + 12 = 12
- 4n = 0 => n = 0 (valid)
5. For 16: 4n + 12 = 16
- 4n = 4 => n = 1 (not valid)
Conclusion
Since n must always be an even integer, the value that cannot be achieved is option C) 8, as it yields an invalid n.
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