Question

daca x+1/x=4 aflati x^2+1/x^2 si x^4+1/x^4

Answer 1

Răspuns:

14

194

Explicație pas cu pas:

• x²+1/x²=(x+1/x)²-2=4²-2=16-2=14

• x^4+1/x^4=(x²+1/x²)²-2=14²-2=196-2=194

Answer 2

$\frac{x + 1}{x} = 4$

$x + 1 = 4x$

$x - 4x = - 1$

$- 3x = - 1$

$x = \frac{ - 1}{ - 3} = \frac{1}{3}$

$( \frac{x + 1}{x} )^{2} = {4}^{2}$

$\frac{ {x}^{2} + 2x + 1 }{ {x}^{2} } = 16$

$\frac{ {x}^{2} + 1}{ {x}^{2} } + \frac{2x}{ {x}^{2} } = 16$

$\frac{2x}{ {x}^{2} } = \frac{2 \times \frac{1}{3} }{( \frac{1}{3})^{2} } = \frac{2}{ \frac{1}{3} } = 6$

$\frac{ {x}^{2} + 1 }{ {x}^{2} } + 6 = 16$

$\frac{ {x}^{2} + 1 }{ {x}^{2} } = 10$

$( \frac{x^{2} + 1 }{x ^{2} } )^{2} = {10}^{2}$

$\frac{ {x}^{4} + 2 {x}^{2} + 1}{ {x}^{4} } = 100$

$\frac{ {x}^{4} + 1 }{ {x}^{4} } + \frac{2x^{2} }{ {x}^{4} } = 100$

$\frac{2 {x}^{2} }{ {x}^{4} } = \frac{2 \times {( \frac{1}{3} )}^{2} }{ {( \frac{1}{3} )}^{4} } = \frac{2}{ {( \frac{1}{3}) }^{2} } = \frac{2}{ \frac{1}{9} } = 18$

$\frac{ {x}^{4} + 1 }{ {x}^{4} } + 18 =100$

$\frac{ {x}^{4} + 1 }{ {x}^{4} } = 82$