Three Homogeneous Quadratics

Algebra Level 4

Given positive real values \( j, k\) and \(l\) such that \[ \begin{array}{l l} j^2 +k^2 + jk & = & 9 \\ k^2 + l^2 + kl & = & 16\\ l^2 + j^2 + lj & = & 25,\\ \end{array} \] \( (j + k + l)^2 \) has the form \( a + b \sqrt{c} \), where \(a, b\) and \(c\) are integers and \( c \) is not a multiple of any square number. What is \( a + b + c\)?

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