Canonical form, rank and signature of a ternary quadratic form

Consider the quadratic form \(f ( \vec { x } ) = 2 x _ { 1 } ^ { 2 } + 4 x _ { 1 } x _ { 2 } + 3 x _ { 2 } ^ { 2 } + 4 x _ { 2 } x _ { 3 } + 5 x _ { 3 } ^ { 2 }\) ,where \(x _ { i } , i = 1 , 2 , 3\) are components of x in \(\mathbb { R } ^ { 3 }\)

1. Reduce \(f ( \vec { x } )\) to the canonical form. Calculate its rang and signature.

2. Write the transition matrix to the basis where f (x) has the canonical form.