The real value of p for which the sum of the squares of the roots of the equation x2-(p-2)x-p-1=o assume least value is___

Asked by Ch | 12th May, 2017, 06:09: PM

Expert Answer:

begin mathsize 16px style straight x squared minus open parentheses straight p minus 2 close parentheses straight x minus straight p minus 1 equals 0
First space we space find space the space value space of space discriminant.
Here comma space straight a equals 1 comma straight b equals negative left parenthesis straight p minus 2 right parenthesis comma space straight c equals negative straight p minus 1
straight b squared minus 4 ac equals open parentheses straight p minus 2 close parentheses squared minus 4 open parentheses negative straight p minus 1 close parentheses
rightwards double arrow straight b squared minus 4 ac equals straight p squared minus 4 straight p plus 4 plus 4 straight p plus 4
rightwards double arrow rightwards double arrow straight b squared minus 4 ac equals straight p squared plus 8
Let comma space straight alpha space and space straight beta space are space the space roots
straight alpha equals fraction numerator straight p minus 2 plus square root of straight p squared plus 8 end root over denominator 2 end fraction space space and space straight beta equals fraction numerator straight p minus 2 minus square root of straight p squared plus 8 end root over denominator 2 end fraction
straight alpha squared plus straight beta squared equals open parentheses fraction numerator straight p minus 2 plus square root of straight p squared plus 8 end root over denominator 2 end fraction close parentheses squared plus open parentheses fraction numerator straight p minus 2 minus square root of straight p squared plus 8 end root over denominator 2 end fraction close parentheses squared
after space solving space this space you space will space get
straight alpha squared plus straight beta squared equals straight p squared minus 2 straight p plus 6
Consider comma space straight f left parenthesis straight p right parenthesis equals straight p squared minus 2 straight p plus 6
straight f apostrophe left parenthesis straight p right parenthesis equals 2 straight p minus 2
To space find space minimum space value space or space maximum space value
straight f apostrophe open parentheses straight p close parentheses equals 0 rightwards double arrow straight p equals 1
straight f apostrophe apostrophe left parenthesis straight p right parenthesis equals 2 greater than 0
rightwards double arrow function space has space minimum space value space at space straight p equals 1.
end style

Answered by Sneha shidid | 12th May, 2017, 07:06: PM