If and are roots of x^2+px+2=0 and 1,1 are roots of 2x^2+2qx+1=0, then (-1)(-1)(+1)(+1) is…

Question

If $\alpha$ and $\beta$ are roots of $x^2+px+2=0$ and $\dfrac1\alpha,\dfrac1\beta$ are roots of $2x^2+2qx+1=0$, then $\left(\alpha-\dfrac1\alpha\right)\left(\beta-\dfrac1\beta\right)\left(\alpha+\dfrac1\beta\right)\left(\beta+\dfrac1\alpha\right)$ is equal to

Accepted Answer

$\alpha\beta=2$, $\alpha+\beta=-p$, and $\dfrac1\alpha+\dfrac1\beta=-q \implies p=2q$.

$$\left(\alpha-\frac1\alpha\right)\left(\beta-\frac1\beta\right)\left(\alpha+\frac1\beta\right)\left(\beta+\frac1\alpha\right)=\left(\alpha\beta+\frac1{\alpha\beta}-\frac\alpha\beta-\frac\beta\alpha\right)\left(\alpha\beta+\frac1{\alpha\beta}+2\right)$$
$$=\left(\frac92-\frac{\alpha^2+\beta^2}2\right)\left(\frac92+2\right)=\frac94[5-(p^2-4)]=\frac94(9-p^2).$$

Solutions · 1