Matematické Fórum

stuart clark · 23. 06. 2016 14:18

If $a,b,c$ are $3$ real numbers and $a^2+ab+b^2=9\;,b^2+bc+c^2=52$ and $c^2+ac+a^2 = 49\;,$ Then $\frac{49b^2+39bc+9c^2}{a^2}=$

sugyman · 26. 06. 2016 00:58

Substracting second and third equation gives us

Also

Therefore

So $b=\frac{2a}{3}$ or $b=\frac{6a}{5}$ Each of these two cases satisfies the three given equations with $c=\frac{-34a}{9}$ and $c=\frac{58a}{15}$ respectively (also $a=\pm\frac{9}{\sqrt{19}}$ and $a=\pm\frac{15}{\sqrt{91}}$ but who cares..).
Finally plugging in these these in our expression, we get $52$ and $\frac{9652}{25}$ .

stuart clark · 01. 07. 2016 10:00

Thanks ↑ sugyman:

Matematické Fórum

#1 23. 06. 2016 14:18

Value of expression

#2 26. 06. 2016 00:58

Re: Value of expression

#3 01. 07. 2016 10:00

Re: Value of expression

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