Question

3. 1 Dacă a/b = 2 - sqrt(2) atunci valoarea raportului (a ^ 2 - 2b ^ 2)/(2ab) este egală cu:

Answer 1

Răspuns:

Explicație pas cu pas:

$\dfrac{a}{b} = 2 - \sqrt{2}$

$\dfrac{a^{2} - 2 b^{2} }{2ab} = \dfrac{a^{2}}{2ab} - \dfrac{2 b^{2} }{2ab} = \dfrac{1}{2} \cdot \dfrac{a}{b} - \dfrac{b}{a} = \dfrac{1}{2} \cdot (2 - \sqrt{2}) - \dfrac{1}{2 - \sqrt{2}} = \dfrac{2 - \sqrt{2}}{2} - \dfrac{2 + \sqrt{2}}{(2 - \sqrt{2})(2 + \sqrt{2})} = \dfrac{2 - \sqrt{2}}{2} - \dfrac{2 + \sqrt{2}}{4 - 2} = \dfrac{2 - \sqrt{2} - 2 - \sqrt{2} }{2} = -\dfrac{2\sqrt{2} }{2} =\bf - \sqrt{2}$

sau:

$\dfrac{a}{b} = 2 - \sqrt{2} \iff a = (2 - \sqrt{2})b$

$\dfrac{(2 - \sqrt{2})^{2}b^{2} - 2 b^{2} }{2(2 - \sqrt{2})b \cdot b} = \dfrac{(4 - 4\sqrt{2} + 2)b^{2} - 2 b^{2} }{2(2 - \sqrt{2})b^{2}} = \dfrac{(4 - 4\sqrt{2} + 2 - 2)b^{2}}{2(2 - \sqrt{2})b^{2}} = \dfrac{4(1 - \sqrt{2})}{2(2 - \sqrt{2})} = \dfrac{2(1 - \sqrt{2})(2 + \sqrt{2})}{(2 - \sqrt{2})(2 + \sqrt{2})} = \dfrac{2(2 + \sqrt{2} - 2\sqrt{2} - 2)}{4 - 2} = \dfrac{2(- \sqrt{2})}{2} = \bf - \sqrt{2}$