Symmetric Inequality Problem

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

To restate the problem,

If xx, yy and zz are real numbers such that: {x+y+z=5xy+yz+zx=3\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 xx, yy and zz 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 xx, yy and zz could be any real numbers, not just positive.

Any help/discussion would be much appreciated!

Note by Arthur Conmy
2 years, 2 months 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 - 2 years, 2 months ago

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Fixed btw

Arthur Conmy - 2 years, 2 months ago

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yup

Vilakshan Gupta - 2 years, 2 months ago

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Thanks Vilakshan. I leant from both your and Sharky's answers.

Arthur Conmy - 2 years, 2 months ago

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actually u have put a dot by mistake in front of 251

Vilakshan Gupta - 2 years, 2 months ago

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okay ! great

Vilakshan Gupta - 2 years, 2 months ago

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Hey buddy is the answer 132\frac{13}{2}?

Sathvik Acharya - 2 years, 2 months ago

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i m getting 133\frac{13}{3} as maximum value and -1 as minimum value

Vilakshan Gupta - 2 years, 2 months ago

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Yep same here! What was your method?

Sathvik Acharya - 2 years, 2 months ago

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but u said u are getting 132\frac{13}{2} . I used the same method which i referred to in the link i gave.

Vilakshan Gupta - 2 years, 2 months ago

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@Vilakshan Gupta Oops! Sorry I am getting 133\frac{13}{3}. I actually did not check the link out. Let me check it now

Sathvik Acharya - 2 years, 2 months ago

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@Sathvik Acharya Ok. Tell me about your method please

Vilakshan Gupta - 2 years, 2 months ago

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@Vilakshan Gupta Cauchy Schwarz inequality

Sathvik Acharya - 2 years, 2 months ago

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@Sathvik Acharya same method as mine then? or in a different way?

Vilakshan Gupta - 2 years, 2 months ago

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