Numbers can change the Perceptions. Part I

Algebra Level 4

$\sqrt{x^2+\sqrt{x^2+1729}}+\sqrt{x^2-\sqrt{x^2+1729}}=2$

Find the number of real values of $x$ satisfying the above equation.

Zero One Two More than two

2 solutions

Personal Data
Jun 6, 2015

It is clear that ${ x }^{ 2 }\ge 0$ so $\min { \left\{ \sqrt { { x }^{ 2 }+\sqrt { { x }^{ 2 }+1729 } } \right\} } =\sqrt [ 4 ]{ 1729 } >2$ at $x=0$ . Now since the second term of $LHS$ is necessarily nonnegative then $LHS>RHS$ so the equation has no solutions.

That's what I call "Elegant Approach". Great Solution. Thanks for posting it.

Sandeep Bhardwaj - 6 years ago

Thank you very much.

Personal Data - 6 years ago

Omg u are a god

genis dude - 4 years, 6 months ago

Chew-Seong Cheong
Jul 3, 2015

${\color{#D61F06}\text{The solution is wrong.}}$

$\sqrt{x^2-\sqrt{x^2+1729}}$ is not real, therefore, the equation has no real solution.

Moderator note:

Great observation!

How do you know if that not real ?

Daniel Sugihantoro - 4 years, 7 months ago

Sorry, my solution is wrong. I thought it was $\sqrt{x^{\color{#D61F06}4}-\sqrt{x^2+1729}}$ . I will delete it after you have read this reply.

Chew-Seong Cheong - 4 years, 7 months ago

Numbers can change the Perceptions. Part I

2 solutions

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