Question

Daca x+1/x=6 calculati x-1/x=
Daca x-1/x=4 calculati x+1/x=

Answer 1

x+1/x=6
(x²+1)/x=6
x²+1=6x
x²-6x+1=0
Δ=36-4=32
x1=(6-4√2)/2=2(3-2√2)/2=3-2√2
x2=3+2√2
S={3-2√2, 3+2√2)
daca x=3-2√2
x-1/x=3-2√2-1/(3-2√2)=3-2√2-(3+2√2)/(9-8)=
=3-2√2-3-2√2=-4√2
daca x=3+2√2
x-1/x=3+2√2-1/(3+2√2)=3+2√2-(3-2√2)/9-8)=
=3+2√2-3+2√2=4√2

x-1/x=4
(x²-1)/x=4
x²-1=4x
x²-4x-1=0
Δ=16+4=20
x1=(4-2√5)/2=2(2-√5)/2=2-√5
x2=2+√5
S={2-√5, 2+√5)
daca x=2-√5
x+1/x=2-√5+1/(2-√5)=2-√5+(2+√5)/(4-5)=
=2-√5-(2+√5)=2-√5-2-√5=-2√5
daca x=2+√5
x+1/x=2+√5+1/(2+√5)=2+√5+(2-√5)/(4-5)=
=2+√5-(2-√5)=2+√5-2+√5=2√5