Respuesta :
The value of x in the equation [tex](x-5)^{2} +3(x-5)+9=0[/tex]is x=(7+3[tex]\sqrt{3}i[/tex])/2,
(7-3[tex]\sqrt{3}i[/tex])/2.
Given equation [tex](x-5)^{2} +3(x-5)+9=0[/tex] .
We have to find the solution of the equation.
Equation is solved to find the value of variables. The number of values of the variable depends on the highest power of variables.
let (x-5)=u-------------------1
equation becomes [tex]u^{2} +3u+9=0[/tex]
solution of a quadratic formula
x=-b±[tex]\sqrt{b^{2} -4ac}[/tex]/2a
on comparing with general form a=1,b=3, c=9
u=-3±[tex]\sqrt{3^{2} -4*1*9} /2[/tex]
=-3±[tex]\sqrt{9-36}/2[/tex]
=-3±[tex]3\sqrt{-27}/2[/tex]
=-3±[tex]3\sqrt{3} i/2[/tex]
[tex]u_{1} =-3[/tex]+3[tex]\sqrt{3}/2[/tex] , [tex]u_{2}[/tex]=-3-3[tex]\sqrt{3}/2[/tex]
put the values of [tex]u_{1}[/tex] in equation 1
x-5=[tex]u_{1}[/tex]
x-5=-3+[tex]3\sqrt{3} /2[/tex]
x=-3+3[tex]\sqrt{3}i/2+5[/tex]
x=7+3[tex]\sqrt{3} i/2[/tex]
put the values of [tex]u_{2}[/tex] in equation2
x-5=[tex]u_{2}[/tex]
x-5=-3-3[tex]\sqrt{3} /2[/tex]
x=7-3[tex]\sqrt{3}/2[/tex]
Hence the values of x are 7+3[tex]\sqrt{3}i/2[/tex], 7-3[tex]\sqrt{3}i/2[/tex].
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