I am struggling with understanding https://nrich.maths.org/251.

To restate the problem,

If \(x\), \(y\) and \(z\) are real numbers such that: \(\begin{cases}x+y+z=5 \\ xy+yz+zx=3\end{cases}\) , What is the largest value that any one of these numbers can have?

In particular, I do not understand the first solution given, and while the second I am getting a grip with (creates a quadratic uses the discriminant inequality since \(x\), \(y\) and \(z\) are real numbers), would like to ask whether any classical inequalities can be used here, as I would be personally more satisfied with this.

The problem I had with applying inequalities I knew was that \(x\), \(y\) and \(z\) could be any real numbers, not just positive.

Any help/discussion would be much appreciated!

## Comments

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TopNewesti m getting \(\frac{13}{3} \) as maximum value and -1 as minimum value – Vilakshan Gupta · 1 month ago

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– Sathvik Acharya · 1 month ago

Yep same here! What was your method?Log in to reply

– Vilakshan Gupta · 1 month ago

but u said u are getting \(\frac{13}{2} \) . I used the same method which i referred to in the link i gave.Log in to reply

– Sathvik Acharya · 1 month ago

Oops! Sorry I am getting \(\frac{13}{3}\). I actually did not check the link out. Let me check it nowLog in to reply

– Vilakshan Gupta · 1 month ago

Ok. Tell me about your method pleaseLog in to reply

– Sathvik Acharya · 1 month ago

Cauchy Schwarz inequalityLog in to reply

– Vilakshan Gupta · 1 month ago

same method as mine then? or in a different way?Log in to reply

Hey buddy is the answer \(\frac{13}{2}\)? – Sathvik Acharya · 1 month ago

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okay ! great – Vilakshan Gupta · 1 month ago

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actually u have put a dot by mistake in front of 251 – Vilakshan Gupta · 1 month ago

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The link u have referred is not opening, it says the page not found.However, u can see my solution to this problem : https://brilliant.org/problems/almost-vietas/#!/solution-comments/171217/ – Vilakshan Gupta · 1 month ago

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– Arthur Conmy · 1 month ago

Thanks Vilakshan. I leant from both your and Sharky's answers.Log in to reply

– Arthur Conmy · 1 month ago

Fixed btwLog in to reply

– Vilakshan Gupta · 1 month ago

yupLog in to reply