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When a=x+(1/x) and b=x-(1/x), what is a^2 – b^2?

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by Max@Math Revolution » Wed May 09, 2018 2:16 am

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Difficulty—

[GMAT math practice question]

When $$a=x+\frac{1}{x}\ and\ b=x-\frac{1}{x}$$ $$what\ is\ a^2-b^2$$

$$A.\ x^2+\frac{1}{^{x^2}}$$
$$B.\ x^2-\frac{1}{x^2}$$
$$C.\ 1$$
$$D.\ 2$$
$$E.\ 4$$
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a and b

by GMATGuruNY » Wed May 09, 2018 2:50 am
Max@Math Revolution wrote:[GMAT math practice question]

When $$a=x+\frac{1}{x}\ and\ b=x-\frac{1}{x}$$ $$what\ is\ a^2-b^2$$

$$A.\ x^2+\frac{1}{^{x^2}}$$
$$B.\ x^2-\frac{1}{x^2}$$
$$C.\ 1$$
$$D.\ 2$$
$$E.\ 4$$
Let x=1, with the following results:
a = 1 + (1/1) = 2.
b = 1 - (1/1) = 0.
a² - b² = 2² - 0² = 4.

When x=1, the correct answer must yield a value of 4.
Only E is viable.

The correct answer is E.
Last edited by GMATGuruNY on Wed May 09, 2018 6:54 am, edited 1 time in total.
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by Brent@GMATPrepNow » Wed May 09, 2018 6:30 am
Max@Math Revolution wrote:[GMAT math practice question]

When $$a=x+\frac{1}{x}\ and\ b=x-\frac{1}{x}$$ $$what\ is\ a^2-b^2$$

$$A.\ x^2+\frac{1}{^{x^2}}$$
$$B.\ x^2-\frac{1}{x^2}$$
$$C.\ 1$$
$$D.\ 2$$
$$E.\ 4$$
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
Brent Hanneson - Creator of GMATPrepNow.com
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by Scott@TargetTestPrep » Thu May 10, 2018 4:51 pm
Max@Math Revolution wrote:[GMAT math practice question]

When $$a=x+\frac{1}{x}\ and\ b=x-\frac{1}{x}$$ $$what\ is\ a^2-b^2$$

$$A.\ x^2+\frac{1}{^{x^2}}$$
$$B.\ x^2-\frac{1}{x^2}$$
$$C.\ 1$$
$$D.\ 2$$
$$E.\ 4$$
Rather than squaring a and b individually, we see that a^2 - b^2 is a difference of squares:

a^2 - b^2

(a + b)(a - b)

(x + 1/x + x - 1/x)(x + 1/x - (x - 1/x))

(2x)(2/x) = 4

Alternate Solution:

a^2 - b^2

(x + 1/x)^2 - (x - 1/x)^2

x^2 + 2 + 1/x^2 - (x^2 - 2 + 1/x^2) = 4

Answer: E

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by Max@Math Revolution » Fri May 11, 2018 12:18 am
=>

a^2 - b^2
= (a+b)(a-b)
= ( x + 1/x + x - 1/x ) ( x + 1/x - ( x - 1/x ) )
= (2x)(2/x)
= 4

Therefore, the answer is E.

Answer : E
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