Q52 / 160 · AP EAMCET 2024 (21 May Shift 2) · Mathematics
The equation $x^4-x^3-6 x^2+4 x+8=0$ has two equal roots. If $\alpha, \beta$ are the other two roots of this equation then $\alpha^2+\beta^2=$
- A4
- Correct answerB5
- C6
- D7
Solution
$x^4-x^3-6 x^2+4 x+8=0$
By Hit and trial $x=2$ is a root of the equation
$\Rightarrow(x-2)\left(x^3+x^2-4 x-4\right)=0$
Again $x=2$ is a root of the equation $\begin{aligned} & \Rightarrow(x-2)^2\left(x^2+3 x+2\right)=0 \\ & \Rightarrow(x-2)^2(x-1)(x+2)=0 \Rightarrow x=2,2,-2,1 \\ & \Rightarrow \alpha=-2, \beta=1 \Rightarrow \alpha^2+\beta^2=5 \end{aligned}$
Again $x=2$ is a root of the equation $\begin{aligned} & \Rightarrow(x-2)^2\left(x^2+3 x+2\right)=0 \\ & \Rightarrow(x-2)^2(x-1)(x+2)=0 \Rightarrow x=2,2,-2,1 \\ & \Rightarrow \alpha=-2, \beta=1 \Rightarrow \alpha^2+\beta^2=5 \end{aligned}$
Hence, the correct answer is option: 'B'.