Mystery Problem of the day

Algebra Level 1

If 1 x + x = 5 \dfrac{1}{x} + x = 5 , then what is 1 x 2 + x 2 \dfrac{1}{x^{2}}+ x^{2} ?

27 25 23 22 24

Uranus Ng by Uranus Ng , 16, Hong Kong

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2 solutions

Square both sides of the equation.

( 1 x + x ) 2 = 5 2 \left(\dfrac{1}{x}+x\right)^2=5^2

1 x 2 + 1 x ( x ) ( 2 ) + x 2 = 25 \dfrac{1}{x^2}+\dfrac{1}{x}(x)(2)+x^2=25

1 x 2 + 2 + x 2 = 25 \dfrac{1}{x^2}+2+x^2=25

1 x 2 + x 2 = 23 \dfrac{1}{x^2}+x^2=\boxed{23}

4 Helpful 0 Interesting 0 Brilliant 1 Confused

I know, you're right. This was just my first post.

Uranus Ng - 3 years, 2 months ago

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You know, just let everyone post their solutions. You don’t need to tell them about that. Upvote their solutions is a good way of saying just that.

Congrats on your first post anyways! Do continue!

Steven Jim - 3 years, 2 months ago
Mahdi Raza
Jun 7, 2020

\[\begin{align} \bigg( x + \dfrac{1}{x}\bigg)^2 &= (5)^2 \\ \\ x^2 + \dfrac{1}{x^2} + 2(\cancel{x})\big(\dfrac{1}{\cancel{x}}\big) &= (5)^2 \\ \\ x^2 + \dfrac{1}{x^2} + 2 &= 25 \\ \\ x^2 + \dfrac{1}{x^2} & = \boxed{23}

\end{align}\]

0 Helpful 0 Interesting 1 Brilliant 0 Confused

The problem poser marked your solution as "brilliant"! :)

Uranus Ng - 1 year ago

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Thank you :)

Mahdi Raza - 1 year ago

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