Given $x^2 - 3ax + a^2 = 0$ and $$\frac{x_1^4-x_2^4}{\sqrt{5}x_1x_2} + x_1 + x_2 -20x_1x_2 - 4 = 0$$ Find $a$.
The answer is $1$ ($a = 1$)
I tried to present $x_1^4 - x_2^4$ as $(x_1+x_2)(x_1-x_2)(x_1+x_2)^2-2x_1x_2$ But I can't replace (x1-x2)
Note: $x_1$ and $x_2$ are the roots of the equation.
By the factorisation, $$(x_1^4-x_2^4)=(x_1+x_2)(x_1-x_2)(x_1^2+x_2^2)$$ We get $$(x_1^4-x_2^4)=(3a)[\sqrt{(x_1+x_2)^2-4x_1x_2}](9a^2-2a^2)$$ $$\implies (x_1^4-x_2^4)=21a^3\sqrt{5}a$$ Can you continue?