Question

if α+β are the roots of the equation x^2+4x+4=0, then find the value of 1+α^2+1+β^2 ; 1+α+1+β ; α+β+β+α ; α^2+β+β^2+α​

Answer 1

Dear Student,

$$\begin{gathered} thegivenequationis{x}^2+4x+4=0 \\ hence\alpha +\beta =-4and\alpha \beta =4 \\ i)1+{\alpha }^2+{\beta }^2+1\Rightarrow 2+{\left(\alpha +\beta \right)}^2-2\alpha \beta \Rightarrow 2+{\left(-4\right)}^2-2\times 4 \\ =2+16-8=10 \\ ii)1+\alpha +1+\beta \Rightarrow 2+\alpha +\beta =2-4\Rightarrow -2 \\ 1+\alpha +1+\beta =-2 \\ iii)\alpha +\beta +\beta +\alpha =2\times \left(\alpha +\beta \right)=2\times \left(-4\right)\Rightarrow -8 \\ iv){\alpha }^2+\beta +{\beta }^2+\alpha \\ =\alpha +\beta +{\alpha }^2+{\beta }^2\Rightarrow \alpha +\beta +{\left(\alpha +\beta \right)}^2-2\alpha \beta \\ =-4+{\left(-4\right)}^2-2\times 4 \\ =-4+16-8\Rightarrow 4 \end{gathered}$$
Regards,