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If the equation $x^4+7 x^3+18 x^2+20 x+8=0$ has a repeated root, then that repeated root is
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Verified Answer
The correct answer is:
$-2$
Given equation is $x^4+7 x^3+18 x^2+20 x+8=0$
$\begin{aligned}
& \Rightarrow x^4+2 x^3+5 x^3+10 x^2+8 x^2+16 x+4 x+8=0 \\
& \Rightarrow(x+2)\left(x^3+5 x^2+8 x+4\right)=0 \\
& \Rightarrow(x+2)\left(x^3+2 x^2+3 x^2+6 x+2 x+4\right)=0 \\
& \Rightarrow(x+2)(x+2)\left(x^2+3 x+2\right)=0
\end{aligned}$
so repeated roots $=-2$
$\begin{aligned}
& \Rightarrow x^4+2 x^3+5 x^3+10 x^2+8 x^2+16 x+4 x+8=0 \\
& \Rightarrow(x+2)\left(x^3+5 x^2+8 x+4\right)=0 \\
& \Rightarrow(x+2)\left(x^3+2 x^2+3 x^2+6 x+2 x+4\right)=0 \\
& \Rightarrow(x+2)(x+2)\left(x^2+3 x+2\right)=0
\end{aligned}$
so repeated roots $=-2$
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