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When a=x+(1/x) and b=x-(1/x), what is a^2 – b^2?
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05 Feb 2021, 09:16
05 Feb 2021, 09:16
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When \(a=x+(\frac{1}{x})\) and \(b=x-(\frac{1}{x})\), what is \(a^2 – b^2\)?
\(A. x^2 + \frac{1}{x^2}\)
\(B. x^2 - \frac{1}{x^2}\)
\(C. 1\)
\(D. 2\)
\(E. 4\)
\(A. x^2 + \frac{1}{x^2}\)
\(B. x^2 - \frac{1}{x^2}\)
\(C. 1\)
\(D. 2\)
\(E. 4\)
Retired Moderator
Joined: 10 Apr 2015
Posts: 6218
Given Kudos: 136
Re: When a=x+(1/x) and b=x-(1/x), what is a^2 – b^2?
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05 Feb 2021, 09:25
05 Feb 2021, 09:25
1
Carcass wrote:
When \(a=x+(\frac{1}{x})\) and \(b=x-(\frac{1}{x})\), what is \(a^2 – b^2\)?
\(A. x^2 + \frac{1}{x^2}\)
\(B. x^2 - \frac{1}{x^2}\)
\(C. 1\)
\(D. 2\)
\(E. 4\)
\(A. x^2 + \frac{1}{x^2}\)
\(B. x^2 - \frac{1}{x^2}\)
\(C. 1\)
\(D. 2\)
\(E. 4\)
Given: a = x + 1/x and b = x - 1/x
Our goal is to find the value of a² - b²
To do so, it's useful to recognize that the expression a² - b² is a difference of squares, which means we can rewrite it.
When we do this, we get: a² - b² = (a + b)(a - b)
Now replace a and b with their equivalent expressions to get: a² - b² = [(x + 1/x) + (x - 1/x)][(x + 1/x) - (x - 1/x)]
Simplify to get: a² - b² = [2x][2/x]
Simplify more to get: a² - b² = 4
Answer: E
Cheers,
Brent
Retired Moderator
Joined: 10 Apr 2015
Posts: 6218
Given Kudos: 136
Re: When a=x+(1/x) and b=x-(1/x), what is a^2 – b^2?
[#permalink]
05 Feb 2021, 09:26
05 Feb 2021, 09:26
2
Carcass wrote:
When \(a=x+(\frac{1}{x})\) and \(b=x-(\frac{1}{x})\), what is \(a^2 – b^2\)?
\(A. x^2 + \frac{1}{x^2}\)
\(B. x^2 - \frac{1}{x^2}\)
\(C. 1\)
\(D. 2\)
\(E. 4\)
\(A. x^2 + \frac{1}{x^2}\)
\(B. x^2 - \frac{1}{x^2}\)
\(C. 1\)
\(D. 2\)
\(E. 4\)
Alternatively, we can see what a² - b² equals for some value of x.
Let's see what happens when x = 1
a = x + 1/x
= 1 + 1/1
= 1 + 1
= 2
So, a² = 4
b = x - 1/x
= 1 - 1/1
= 1 - 1
= 0
So, b² = 0
So, when x = 1, a² - b² = 4 - 0 = 4
In other words, when we INPUT x = 1, the expression a² - b² = 4
Now let's check the answer choices to see which one yields an output of 4 when we input x = 1
A. x² + 1/x² = 1² + 1/1² = 1 + 1 = 2. We want 4, so ELIMINATE A
B. x² - 1/x² = 1² - 1/1² = 1 - 1 = 0. We want 4, so ELIMINATE B
C. 1. We want 4, so ELIMINATE C
D. 2. We want 4, so ELIMINATE D
E. 4. PERFECT!!!
Answer: E
Cheers,
Brent
