How many complex roots are there in #x^3=8#? Your helping hand is highly appreciated. Thanks

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Answer 1 · 2018-05-10T04:08:50

#(1)x=2...toreal # root

#(2)x=-1+sqrt3ior x=-1-sqrt3ito # two complex roots.

We know that,

#color(green)((A)a^3-b^3=(a-b)(a^2+ab+b^2)#

Here,

#x^3=8#

#=>x^3-8=0#

#=>x^3-2^3=0...tocolor(green)(Apply(A)#

#=>(x-2)(x^2+2x+4)=0#

#=>x-2=0 or x^2+2x+4=0#

#color(red)((1)x=2 or #

#(2)x^2+2x+4=0#

#=>x^2+2x+1+3=0#

#=>x^2+2x+1=-3#

#=>(x+1)^2=3i^2...tocolor(blue)([i^2=-1]#

#=>(x+1)^2=(sqrt3i)^2#

#=>x+1=+-sqrt3i#

#=>x=-1+-sqrt3i#

#=>color(red)(x=-1+sqrt3i or x=-1-sqrt3i#

Hence,

#(1)x=2...toreal # root

#(2)#two complex conjugate roots.

#x=-1+sqrt3ior x=-1-sqrt3ito #