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# if equation x^4+px^3+qx^2+rx+5=0 has four real roots then find the minimum value of pr                                                                 ans.80

Arun Kumar IIT Delhi
6 years ago
Hi
Thanks & Regards, Arun Kumar, Btech,IIT Delhi, Askiitians Faculty
Arun Kumar IIT Delhi
6 years ago
Arun Kumar IIT Delhi
6 years ago
Hello
$\\ \\ \sum r_i=-p \\ \sum r_i.r_j=q \\ \sum r_i.r_jr_k=-r \\ r_i.r_jr_k.r_l=5 \\pr=\sum r_i * \sum r_i.r_jr_k=20+ \\3 terms of r_i^2*product of other any two \\+3 terms of r_j^2product of other any two \\+3 terms of r_k^2product of other any two \\+3 terms of r_l^2product of other any two \\r_i^2r_jr_k={ r_i*r_i.r_jr_k.r_l \over r_l}={5 r_i \over r_j} \\=>pr=20+k \\=>k=12*5( {\sum {r_i \over r_j}}) \\=>({\sum {r_i \over r_j}}) \geq \sqrt{\prod {{\sum {r_i \over r_j}}}}=1 \\=> total sum \geq 80 \\$
Thanks & Regards
Arun Kumar
Btech, IIT Delhi