Question

$\frac{a}{b} = 4 - \sqrt{2} \: calculati \: \frac{ {a}^{2} + {b}^{2} }{ab }$


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Answer 1

Răspuns:

$\frac{60 - 13\sqrt{2}}{14}$

Explicație pas cu pas:

$\frac{a}{b} = 4 - \sqrt{2}\implies \frac{b}{a} = \frac{1}{4-\sqrt{2}} = \frac{4+\sqrt{2}}{16-2} = \frac{4+\sqrt{2}}{14}\\ \\ \frac{a^2+b^2}{ab} = \frac{a^2}{ab} + \frac{b^2}{ab} = \frac{a}{b} + \frac{b}{a} = 4 - \sqrt{2} + \frac{4+\sqrt{2}}{14} = \frac{14\cdot 4 - 14\sqrt{2} + 4 + \sqrt{2}}{14} = \frac{15\cdot 4 - 13\sqrt{2}}{14} = \frac{60 - 13\sqrt{2}}{14}$

Answer 2

Răspuns:

Explicație pas cu pas:

(a^2 + b^2)ab = a^2/ab + b^2/ab = a/b + b/a

b/a = 1/(4 - √2) = (4 + √2)/(4 - √2)(4 + √2) = (4 + √2)/(16 - 2) = (4+ √2)/14

a/b + b/a = 4 - √2 + (4 +√2)/14 = ((56 - 14√2 + 4 + √2)/14 = (60 -13√2)/14