Which of the following quadratic equations whose real roots x₁, x₂ satisfy the conditions x₁^2+x₂^2=5 ; 3 (x₁^5+x₂^5 )=11 (x₁^3+x₂^3 ) ?

Step-by-step solution

x₁^5+x₂^5 x₁^3+x₂^3 = 11 3 aligned & (x₁^2+x₂^2 ) (x₁^3+x₂^3 )-x₁^2 x₂^2 (x₁+x₂ ) x₁^3+x₂^3 = 11 3 & [ a^5+b^5= (a^2+b^2 ) (a^3+b^3 )-a^2 b^2(a+b) ] aligned (x₁^2+x₂^2 )- x₁^2 x₂^2 (x₁+x₂ ) (x₁+x₂ ) (x₁^2-x₁ x₂+x₂^2 ) = 11 3 5- x₁^2 x₂^2 5-x₁ x₂ = 11 3 Let x₁ x₂=t array ll & 5- t^2 5-t = 11 3 & 3 (25-5 t-t^2 )=55-11 t array aligned & & 3 t^2+4 t-20 & =0 & & t & =2,- 10 3 & & x₁ x₂ & =2, -10 3 aligned When x₁ x₂=2 (x₁+x₂ )^2=x₁^2+2 x₁ x₂+x₂^2=5+2 2=9 x₁+x₂= 3 When x₁ x₂=- 10 3 (x₁+x₂ )^2=5- -20 3 =- 5 3 < 0 which is