Let α, β be the roots of the equation x^2 - ax - b = 0 with Im(α) < Im(β).

Question

Let \(\alpha, \beta\) be the roots of the equation \(x^{2}-a x-b=0\) with \(\operatorname{Im}(\alpha)<\operatorname{Im}(\beta).\) Let \(P_{n}=\alpha^{n}-\beta^{n}.\) If \(P_{3}=-5 \sqrt{7} i, \ P_{4}=-3 \sqrt{7} i, \ P_{5}=11 \sqrt{7} i\) and \(P_{6}=45 \sqrt{7} i,\) then \(\left|\alpha^{4}+\beta^{4}\right|\) is equal to _______.

Answer 1

Answer is: 31 

\(\alpha+\beta=\mathrm{a} \quad \alpha \beta=-\mathrm{b}\)

\(\mathrm{P}_{6}=\mathrm{aP}_{5}+\mathrm{bP}_{4}\)

\(45 \sqrt{7} \mathrm{i}=\mathrm{a} \times 11 \sqrt{7} \mathrm{i}+\mathrm{b}(-3 \sqrt{7}) \mathrm{i}\) 

\(45=11 \mathrm{a}-3 \mathrm{~b}\)  ...(1)

and

\(\mathrm{P}_{5}=\mathrm{aP}_{4}+\mathrm{bP}_{3}\)

\(11 \sqrt{7} \mathrm{i}=\mathrm{a}(-3 \sqrt{7} \mathrm{i})+\mathrm{b}(-5 \sqrt{7} \mathrm{i})\)

\(11=-3 \mathrm{a}-5 \mathrm{~b}\)    ...(2)

\(\mathrm{a}=3, \mathrm{~b}=-4\)

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

\(=\sqrt{-63+4.4^{4}}\)

\(=\sqrt{-63+1024}=\sqrt{961}=31\)