[tex]1) {( \sqrt{3} + \sqrt{2} )}^{ - 2} [/tex]
[tex] = \frac{1}{( \sqrt{3} + \sqrt{2}) ^{2} } [/tex]
[tex] = \frac{1}{3 + 2 \sqrt{6} + 2} [/tex]
[tex] = \frac{1}{5 + 2 \sqrt{6} } [/tex]
[tex]2) \frac{x + 1}{x} = \frac{3}{2} [/tex]
[tex]2(x + 1) = 3x[/tex]
[tex]2x + 2 = 3x[/tex]
[tex]2x - 3x = - 2[/tex]
[tex] - x = - 2 \: | \times ( - 1)[/tex]
[tex]x = 2[/tex]
[tex] \frac{ {x}^{2} + 1}{ {x}^{2} } = \frac{ {2}^{2} + 1}{ {2}^{2} } = \frac{4 + 1}{4} = \frac{5}{4} [/tex]
[tex]3) \frac{x - 1}{x} = 5 \sqrt{2} [/tex]
[tex]5 \sqrt{2} x = x - 1 \: |( \: )^{2} [/tex]
[tex] {(5 \sqrt{2} x)}^{2} = {(x - 1)}^{2} [/tex]
[tex] {50x}^{2} = {x}^{2} - 2x + 1[/tex]
[tex]50 {x}^{2} - {x}^{2} + 2x - 1 = 0[/tex]
[tex]49 {x}^{2} + 2x - 1 = 0[/tex]
[tex]a = 49[/tex]
[tex]b = 2[/tex]
[tex]c = - 1[/tex]
[tex]\Delta = {b}^{2} - 4ac[/tex]
[tex]\Delta = {2}^{2} - 4 \times 49 \times ( - 1)[/tex]
[tex]\Delta = 4 + 196[/tex]
[tex]\Delta = 200[/tex]
[tex]x_{1,2} = \frac{ - b\pm \sqrt{\Delta} }{2a} [/tex]
[tex]x_{1,2} = \frac{ - 2 \pm \sqrt{200}}{2 \times 49}[/tex]
[tex]x_{1,2}= \frac{ - 2\pm10 \sqrt{2} }{98} [/tex]
[tex]x_{1}= \frac{ - 2 + 10 \sqrt{2} }{98} [/tex]
[tex]x_{1} = \frac{2( - 1 + 5 \sqrt{2} )}{98} [/tex]
[tex]x_{1} = \frac{ - 1 + 5 \sqrt{2} }{49} [/tex]
[tex]x_{2} = \frac{ - 2 - 10 \sqrt{2} }{98} [/tex]
[tex]x_{2} = \frac{2( - 1 - 5 \sqrt{2} )}{98} [/tex]
[tex]x_{2} = \frac{ - 1 - 5 \sqrt{2} }{49} [/tex]
