Value of this expression

Question

If $\alpha$ and $\beta$ are the roots of the equation $$x^2 + x − 3 = 0$$ find the value of the expression $4\beta^2 − \alpha^3$.

I tried using sum of roots and product of roots formulas but could not get the answer.

Answer 1

We have $$\beta^2=3-\beta\\4\beta^2=12-4\beta$$ Also $$\alpha^2=3-\alpha\\\alpha^3=3\alpha-\alpha^2=3\alpha-3+\alpha=4\alpha-3$$ Thus, $$4\beta^2-\alpha^3=15-4(\alpha+\beta)=15+4=19$$