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# Help Me!

Find $$ab + bc + ca$$ given that:

$$a^{2} + ab + b^{2} = 2$$

$$b^{2} + bc + c^{2} = 1$$

$$c^{2} + ca + a^{2} = 3$$

Giveb that a, b, and c are positive numbers.

Note by Christian Daang
1 year, 3 months ago

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$$a^2+ab+b^2=2....Eq.1$$

$$b^2+bc+c^2=1....Eq.2$$

$$c^2+ac+a^2=3....Eq.3$$

Eq.1-Eq.2

$$(a+b+c)(a-c)=1...Eq.4$$

Eq.3-Eq.2

$$(a+b+c)(a-b)=1...Eq.5$$

Eq.3-Eq.1

$$(a+b+c)(c-b)=1...Eq.6$$

Comparing Eq.4,5,6, we get $$a=b=c$$

Substituting these values we get an inconsistent system and hence we conclude that there is no real solution. · 1 year, 3 months ago

Is the question correct? There don't exist positive $$a,b,c$$ for which these equations are true. · 1 year, 3 months ago