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09 May 2024, 01:30
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Difficulty:
45%
(medium)
Question Stats:
70% (01:47) correct
30%
(01:57)
wrong
based on 279
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If \(a\) and \(b\) are integers such that \(a < b < 0\) and \(a^2 = b^2 + 7\), what is the value of \(a-b\)?
A. -7
B. -4
C. -3
D. -1
E. 1
A. -7
B. -4
C. -3
D. -1
E. 1
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01 Jun 2024, 04:45
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Official Solution:
If \(a\) and \(b\) are integers such that \(a < b < 0\) and \(a^2 = b^2 + 7\), what is the value of \(a-b\)?
A. -7
B. -4
C. -3
D. -1
E. 1
Given: \(a^2=b^2+7\).
Rearrange and apply \(a^2 - b^2 = (a - b)(a + b)\) to get \((a - b)(a + b) = 7\). Since \(a\) and \(b\) are integers, both \(a + b\) and \(a - b\) are integers as well. Thus, we have the product of two integers equal to 7. There are only two combinations of such factors possible: (1, 7) and (-1, -7). Given that both \(a\) and \(b\) are negative, the first case is not possible, so \(a - b\) must be either -1 or -7. However, it cannot be -7 because in this case, \(a + b\) must be -1, and no two negative integers yield a sum of -1. Therefore, \(a - b = -1\).
Answer: D
If \(a\) and \(b\) are integers such that \(a < b < 0\) and \(a^2 = b^2 + 7\), what is the value of \(a-b\)?
A. -7
B. -4
C. -3
D. -1
E. 1
Given: \(a^2=b^2+7\).
Rearrange and apply \(a^2 - b^2 = (a - b)(a + b)\) to get \((a - b)(a + b) = 7\). Since \(a\) and \(b\) are integers, both \(a + b\) and \(a - b\) are integers as well. Thus, we have the product of two integers equal to 7. There are only two combinations of such factors possible: (1, 7) and (-1, -7). Given that both \(a\) and \(b\) are negative, the first case is not possible, so \(a - b\) must be either -1 or -7. However, it cannot be -7 because in this case, \(a + b\) must be -1, and no two negative integers yield a sum of -1. Therefore, \(a - b = -1\).
Answer: D
General Discussion
09 May 2024, 01:51
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Bunuel
On a number-line,
------ a -------- b ------- 0 ---------
Inference: Both \(a\) and \(b\) are negative.
\(a^2 = b^2 + 7\)
\(a^2 - b^2 = 7\)
\((a+b)(a-b) = 7\)
The following possibilities hold:
| (a+b) | (a-b) | (a+b)*(a-b) | Comments |
|---|---|---|---|
| 1 | 7 | 7 | Invalid as (a+b) should be negative |
| 7 | 1 | 7 | Invalid as (a+b) should be negative |
| -1 | -7 | 7 | Invalid as (a+b) should be less than -1 |
| -7 | -1 | 7 | Valid |
Out of the above 4 possibilities, only one of them is valid.
Hence, (a-b) = -1
Option D
