MHB Minimum of $a$ for $a^2+2b^2+c^2+ab-bc-ac=1$

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The problem involves finding the minimum value of the variable $a$ given the equation $a^2 + 2b^2 + c^2 + ab - bc - ac = 1$. Participants discuss various approaches to manipulate the equation and apply optimization techniques. The hint suggests considering specific values or constraints for $b$ and $c$ to simplify the problem. Several mathematical strategies, including completing the square and using Lagrange multipliers, are proposed to derive the minimum. Ultimately, the goal is to establish the conditions under which $a$ reaches its minimum while satisfying the given equation.
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Let $a,\,b,\,c$ be real numbers such that $a^2+2b^2+c^2+ab-bc-ac=1$. Find the minimum possible value of $a$.
 
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Hint:

Discriminant of quadratic equation helps...:)
 

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