Wednesday, 13 April 2016

algebra precalculus - sum of all non real roots of the equation in a bi-quadratic equation





Consider the equation 8x416x3+16x28x+a=0(aR), Then the sum of



all non real roots of the equation can be




OPTIONS::(a)1(b)2(c)12(d)None



MyTry:: Let f(x)=8x416x3+16x28x+a, Then f(x)=32x348x2+32x8



And f(x)=96x296x+32=96[x2x+13]=96[(x12)2+112]>0xR




So Using LMVT, We get f(x)=0 has at most 1 real roots and



f(x)=0 has at most 2 real roots



Now How can i solve it after that, Help me



Thanks


Answer





Now How can i solve it after that




Noting that f(1/2)=0 is a key.



We have
f(x)=8(4x36x2+4x1)=8(2x1)(2(x12)2+12)
So, f(x) is decreasing for x<1/2 and is increasing for x>1/2.



By the way,

f(12+s)=8s4+4s232+a
and so f(1/2α)=0 is equivalent to f(1/2+α)=0.



Also, by Vieta's formula, the sum of roots is (16)/8=2.



If a3/2, then f(1/2)0, and the answer is
2(1/2α)(1/2+α)=1



If a>3/2, then f(1/2)>0, and the answer is 2.


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