Let a∈R and let α,β be the roots of the equation x2+601/4x+a=0. If α4+β4=−30, then the product of all possible values of a is \_\_\_\_\_\_\_\_.
Answer: 45
Solutions · 1
Official solution
\[\alpha+\beta=-60^{1/4}, \qquad \alpha\beta=a.\]
\[\alpha^4+\beta^4 = (\alpha^2+\beta^2)^2-2\alpha^2\beta^2 = \big[(\alpha+\beta)^2-2\alpha\beta\big]^2-2a^2 = \big(\sqrt{60}-2a\big)^2-2a^2.\]
Setting this equal to −30:
\[60-4a\sqrt{60}+4a^2-2a^2=-30 \implies 2a^2-4\sqrt{60}\,a+90=0.\]
By Vieta's on this quadratic in a, the product of the two possible values of a is 290=45.