1 Quadratic Equations
Let p and q be real numbers such that $$p \neq 0, p^{3} \neq q$$ and $$p^{3} \neq-q .$$ If $$\alpha$$ and $$\beta$$ are nonzero complex numbers satisfying $$\alpha+\beta=-p$$ and $$\alpha^{3}+\beta^{3}=q$$, then a quadratic equation having $$\frac{\alpha}{\beta}$$ and $$\frac{\beta}{\alpha}$$ as its roots is
1) $$\left(p^{3}+q\right) x^{2}-\left(p^{3}+2 q\right) x+\left(p^{3}+q\right)=0$$
2)$$\left(\mathrm{p}^{3}+\mathrm{q}\right) \mathrm{x}^{2}-\left(\mathrm{p}^{3}-2 \mathrm{q}\right) \mathrm{x}+\left(\mathrm{p}^{3}+\mathrm{q}\right)=0$$
3)$$\left(\mathrm{p}^{3}-\mathrm{q}\right) \mathrm{x}^{2}-\left(5 \mathrm{p}^{3}-2 \mathrm{q}\right) \mathrm{x}+\left(\mathrm{p}^{3}-\mathrm{q}\right)=0$$
4)$$\left(p^{3}-q\right) x^{2}-\left(5 p^{3}+2 q\right) x+\left(p^{3}-q\right)=0$$
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