Răspuns:
a) ; b) a = -2; b = 3
Explicație pas cu pas:
a)
[tex] \frac{ \sqrt{2} }{2} = \frac{1}{\sqrt{2} } \\ [/tex]
[tex]{a}^{2} + a + \frac{1}{a} + \frac{1}{ {a}^{2} } = \Big( \frac{1}{\sqrt{2}} \Big)^{2} + \frac{1}{\sqrt{2}} + \frac{1}{\frac{1}{\sqrt{2}}} + \frac{1}{\Big( \frac{1}{\sqrt{2}} \Big)^{2}} = \\ = \frac{1}{2} + \frac{1}{\sqrt{2}} + \sqrt{2} + 2 = \frac{1 + \sqrt{2} + 2 \sqrt{2} + 4}{2} = \bf \frac{5 + 3 \sqrt{2} }{2} [/tex]
b)
[tex]\sqrt{30 - 12 \sqrt{6} } = a \sqrt{3} + b \sqrt{2}[/tex]
[tex]\sqrt{18 + 12 - 12 \sqrt{6} } = a \sqrt{3} + b \sqrt{2} \\[/tex]
[tex]\sqrt{ {(3 \sqrt{2} )}^{2} - 2 \cdot 6 \sqrt{6} + {(2 \sqrt{3} )}^{2} } = a \sqrt{3} + b \sqrt{2} \\ [/tex]
[tex]\sqrt{ {(3 \sqrt{2} - 2 \sqrt{3} )}^{2} } = a \sqrt{3} + b \sqrt{2} \\ [/tex]
[tex]|3 \sqrt{2} - 2 \sqrt{3}| = a \sqrt{3} + b \sqrt{2} \\ [/tex]
[tex]3 \sqrt{2} \geqslant 2 \sqrt{3}[/tex]
[tex]3 \sqrt{2} - 2 \sqrt{3} = a \sqrt{3} + b \sqrt{2} \\ - 2 \sqrt{3} + 3 \sqrt{2} = a \sqrt{3} + b \sqrt{2}[/tex]
[tex]\implies \bf a = -2 \ ; \ b = 3[/tex]
