Question

Fie ecuatia x^2-2x-4=0 cu radacinile reale x1,x2. Fara a calcula x1,x2, determinati o ecuatie de gradul al 2 lea in y cu radacinile reale y1,y2 care indeplineac conditiile:
a) y1= x1-1 ; y2=x2-1
b)y1=x1/x2; y2=x2/x1

Answer 1

y²-(y1+y2)y+y1y2=0

y²-(x1+x2-2)y+(x1x2-x1-x2+1)=0

dar
x1+x2=-(-2)/1=2
si x1x2=-4
atunci
y²-(2-2)y+(-4-2+1)=0
y²-5=0



y²-(y1+y2)y+y1y2=0

y²-((x1²+x2²)/x1x2) *y +1=0

dar x1²+x2²=(x1+x2)²-2x1x2=4-2*(-4)=4+8=12

deci
y²-(12/(-4) )*y+1=0
y²+3y+1=0

fac o verificare la prima pt ca a rezultat
x1.2= (2+-√(4+16))/2=(2+-2√5)/2=(1+-√5)
y1,2=+-√5 , e bine