What is the number of REAL solutions of the following equation?

Question

#(9/10)^k=-3+x-x^2# where #k# belongs to the set of real numbers?

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Answer 1 · 2018-05-17T11:52:03

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First off, the graph of #a^x, a>0# will be continuous from #-ooto+oo# and will always be positive.

Now we need to know if #-3+x-x^2>=0#

#f(x)=-3+x-x^2#

#f'(x)=1-2x=0#

#x=1/2#

#f#''#(x)=-2<-# so the point at #x=1/2# is a maximum.

#f(1/2)=-3+1/2-(1/2)^2=-11/4#

#-3+x-x^2# is always negative while #(9/10)^x# is always positive, they will never cross and so have no real solutions.