Let $\alpha, \beta$ be the roots of $ax^2+bx+c=0$ with $a>0$ where $a, b, c\in \mathbb{R}$. We need to find an equation which roots are $\alpha, -\beta$.
My Trial:
Now $\alpha+\beta=-\frac{b}{a}$ and $\alpha\beta=\frac{c}{a}$. Since $(\alpha-\beta)^2=(\alpha+\beta)^2-4\alpha\beta=\frac{b^2}{a^2}-4\frac{c}{a}$ therefore $\alpha-\beta=\pm \sqrt{\frac{b^2-4ac}{a^2}}$ and $\alpha(-\beta)=-\frac{c}{a}$.
hence the required equation is $x^2\pm \sqrt{\frac{b^2-4ac}{a^2}}x-\frac{c}{a}=0$ i.e. $ax^2\pm \sqrt{b^2-4ac}x-c=0$.
I got stuck here. I do not know how to identify which sign is appropriate here. Please help.