Let α, β be the roots of the equation x^2 + 2√2x - 1 = 0 . The quadratic equation, whose roots

Question

Let \(\alpha, \beta\) be the roots of the equation \(x^{2}+2 \sqrt{2} x-1=0\). The quadratic equation, whose roots are \( \alpha^{4}+\beta^{4}\) and \(\frac{1}{10}\left(\alpha^{6}+\beta^{6}\right)\), is :

(1) \(x^{2}-190 x+9466=0\)

(2) \(x^{2}-195 x+9466=0\)

(3) \(x^{2}-195 x+9506=0\)

(4) \( x^{2}-180 x+9506=0\) 

Answer 1

Correct option is : (3) \(x^{2}-195 x+9506=0\) 

\( x^{2}+2 \sqrt{2} \ x-1=0\)

\(\alpha+\beta=-2 \sqrt{2}\)

\(\alpha \beta=-1\)

\(\alpha^{4}+\beta^{4}=\left(\alpha^{2}+\beta^{2}\right)^{2}-2 \alpha^{2} \beta^{2}\)

\(=\left((\alpha+\beta)^{2}-2 \alpha \beta\right)^{2}-2(\alpha \beta)^{2}\)

\(=(8+2)^{2}-2(-1)^{2}\)

\(=100-2=98\)

\(\alpha^{6}+\beta^{6}=\left(\alpha^{3}+\beta^{3}\right)^{2}-2 \alpha^{3} \beta^{3}\)

\(=\left((\alpha+\beta)\left((\alpha+\beta)^{2}-3 \alpha \beta\right)^{2}-2(\alpha \beta)^{3}\right.\)

\(=(-2 \sqrt{2}(8+3))^{2}+2\)

\(=(8)(121)+2=970\)

\(\frac{1}{10}\left(\alpha^{6}+\beta^{6}\right)=97\)

\(\mathrm{x}^{2}-(98+97) \mathrm{x}+(98)(97)=0\)

\(\Rightarrow \mathrm{x}^{2}-195 \mathrm{x}+9506=0\)