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  4. The sides of a triangle are in the ratio 1: √3: 2. Then the angles are in the ratio

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Q. The sides of a triangle are in the ratio $1: \sqrt{3}: 2$. Then the angles are in the ratio

TS EAMCET 2017

Solution:

We have,
$(x-1)^{3}+64 =0 $
$\Rightarrow (x-1)^{3} =-64 $
$\Rightarrow (x-1)^{3} =(-4)^{3}$
$\Rightarrow x-1 =-4,-4\, w,-4 \,w^{2} $
$\Rightarrow X =-3,-4\, w+1,-4\, w^{2}+1$
Complex roots of the equations are
$-4\, w+1,-4\, w^{2}+1$
Sum of complex roots are
$-4 \,w+1-4\, w^{2}+1=-4\left(w+w^{2}\right)+2$
$=-4(-1)+2=4+2=6 [\because 1+w+w^{2}=0]$