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If x^2 + 9/x^2 = 31, what is the value of x - 3/x?
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16 Nov 2021, 05:26
16 Nov 2021, 05:26
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If \(x^2 + \frac{9}{x^2} = 31\), what is the value of \(x - \frac{3}{x}\)?
A. 36
B. 25
C. 9
D. 5
E. 3
A. 36
B. 25
C. 9
D. 5
E. 3
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Re: If x^2 + 9/x^2 = 31, what is the value of x - 3/x?
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16 Nov 2021, 06:00
16 Nov 2021, 06:00
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Carcass wrote:
If \(x^2 + \frac{9}{x^2} = 31\), what is the value of \(x - \frac{3}{x}\)?
A. 36
B. 25
C. 9
D. 5
E. 3
A. 36
B. 25
C. 9
D. 5
E. 3
Let \(k = x - \frac{3}{x}\) [Our goal will be to find the value of k]
Square both sides to get: \(k^2 = (x - \frac{3}{x})^2\)
Use FOIL to expand and simplify the right side: \(k^2 = x^2 - 6 + \frac{9}{x^2}\)
Rewrite the right side as follows: \(k^2 = (x^2 + \frac{9}{x^2}) - 6\)
Since it's given that \(x^2 + \frac{9}{x^2} = 31\), we can substitute as follows: \(k^2 = (31) - 6\)
Simplify: \(k^2 = 25\)
So, \(k = 5\) or \(k = -5\)
Check the answer choices.... The correct answer must be D
Retired Moderator
Joined: 09 Nov 2021
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If x^2 + 9/x^2 = 31, what is the value of x - 3/x?
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16 Nov 2021, 08:23
16 Nov 2021, 08:23
1
This one certainly isn't immediately obvious. The first thing to note is that your x^2 + 9/x^2 sure looks like a quadratic, and in fact it looks quite a bit like (x - 3/x)^2 squared. Let's take these suppositions and get to work.
First, let's just square (x - 3/x) to see what happens: by binomial expansion, we see that (x - 3/x)^2 = x^2 - 6 + 9/x^2.
Now this is really close. If we simply add six to it, we get the original x^2 + 9/x^2 statement--this is the hard part to notice! Let's see what happens:
x^2 - 6 + 9/x^2 + 6 = 31
Now let's substitute in the bracketed term to see what happens--this is another way to write exactly the same thing:
(x - 3/x)^2 + 6 = 31
From here, it becomes clear that we can just move the six to the right side:
(x - 3/x)^2 = 25
And from here, we can simply take the root of each side. While the technical \sqrt{25} will be +5 or -5, note that we don't have the -5 option.
The answer is D.
First, let's just square (x - 3/x) to see what happens: by binomial expansion, we see that (x - 3/x)^2 = x^2 - 6 + 9/x^2.
Now this is really close. If we simply add six to it, we get the original x^2 + 9/x^2 statement--this is the hard part to notice! Let's see what happens:
x^2 - 6 + 9/x^2 + 6 = 31
Now let's substitute in the bracketed term to see what happens--this is another way to write exactly the same thing:
(x - 3/x)^2 + 6 = 31
From here, it becomes clear that we can just move the six to the right side:
(x - 3/x)^2 = 25
And from here, we can simply take the root of each side. While the technical \sqrt{25} will be +5 or -5, note that we don't have the -5 option.
The answer is D.
