[tex]3) \\ \\ a + \frac{1}{a} = 4 \: | ^{2} \\ \\ (a + \frac{1}{a})^{2} = {4}^{2} \\ \\ {a}^{2} + 2 \times a \times \frac{1}{a} + \frac{1}{ {a}^{2} } = 16 \\ \\ {a}^{2} + 2 + \frac{1}{ {a}^{2} } = 16 \\ \\ {a}^{2} + \frac{1}{ {a}^{2} } = 14 \\ \\ 5) \\ \\ {4x}^{2} + {9y}^{2} - 4x + 12y + 5 = {4x}^{2} - 4x + 1 + {9y}^{2} + 12y + 4 = (( {2x})^{2} - 2 \times 2x + 1) + ((3y)^{2} + 2 \times 3y \times 2 + {2}^{2}) = (2x - 1)^{2} + (3y + 2)^{2} = 0 \\ \\ Cum \: (2x - 1)^{2},(3y + 2)≥0,Oricare \: ar \: fi \: x,y \: din \: R = > 2x - 1 = 3y + 2 = 0 \: de \: unde \: x = \frac{1}{2} \: si \: y = - \frac{2}{3} [/tex]
