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If x+(1/x)=4, what is the value of x^2+(1/x^2)?

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by Max@Math Revolution » Fri Oct 12, 2018 7:28 am

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[Math Revolution GMAT math practice question]

If x+(1/x)=4, what is the value of x^2+(1/x^2)?

A. 4
B. 10
C. 12
D. 14
E. 16
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x + 1/x

by GMATGuruNY » Fri Oct 12, 2018 7:51 am
Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

If x+(1/x)=4, what is the value of x^2+(1/x^2)?

A. 4
B. 10
C. 12
D. 14
E. 16
[x + (1/x)]² = 4²

x² + 1/x² + 2(x)(1/x) = 16

x² + 1/x² + 2 = 16

x² + 1/x² = 14.

The correct answer is D.
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If x+(1/x)=4, what is the value of x^2+(1/x^2)?

by fskilnik@GMATH » Fri Oct 12, 2018 11:36 am
Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

If x+(1/x)=4, what is the value of x^2+(1/x^2)?

A. 4
B. 10
C. 12
D. 14
E. 16
FOCUS:
$$? = {x^2} + {1 \over {{x^2}}}$$
DATA:
$$x + {1 \over x} = 4\,\,\,\,\left( * \right)$$
FOCUS-DATA CONNECTION:
$$?\,\, = \,\,{x^2} + \underline 2 + {1 \over {{x^2}}} - \underline 2 \,\, = \,\,{\left( {x + {1 \over x}} \right)^2} - 2\,\,\,\mathop = \limits^{\left( * \right)} \,\,\,\,{4^2} - 2 = 14$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br
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by Max@Math Revolution » Sun Oct 14, 2018 5:45 pm
=>
Since (a+b)^2=a^2+b^2+2ab, a^2+b^2=(a+b)^2-2ab, and
x^2 + 1/x^2 = ( x + 1/x )^2 - 2*x*(1/x) = 4^2 - 2 = 16 -2 = 14.

Therefore, the answer is D.
Answer: D
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by Scott@TargetTestPrep » Tue Oct 16, 2018 5:17 pm
Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

If x+(1/x)=4, what is the value of x^2+(1/x^2)?

A. 4
B. 10
C. 12
D. 14
E. 16
We square both sides of the equation and obtain:

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

x^2 + 2x(1/x) + (1/x)^2 = 16

x^2 + 2 + 1/x^2 = 16

x^2 + 1/x^2 = 14

Answer: D

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