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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 p.r

If equation x^4+px^3+qx^2+rx+5=0 has four real roots then find the minimum value of p.r

Grade:12

1 Answers

Arun Kumar IIT Delhi
askIITians Faculty 256 Points
7 years ago
Hello Student
\\ \\ \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 $r_j^2$product of other any two$ \\+$3 terms of $r_k^2$product of other any two$ \\+$3 terms of $r_l^2$product 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
Askiitians Faculty

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