<s>[INST] Quadratic equation - $\alpha$ and $\beta$ Roots If $\alpha$ and $\beta$ are the roots of the equation $x^2 + 8x - 5 = 0$, find the quadratic equation whose roots are $\frac{\alpha}{\beta}$ and $\frac{\beta}{\alpha}$.
My working out so far:
I know that $\alpha + \beta = -8$ and $\alpha \beta = -5$ (from the roots) and thenIi go on to work out that
$\alpha= -8-\beta$ and $\beta= -8-\alpha$, then I substitute into what the question asks me.
$\frac{-8-\beta}{-8-\alpha}$ and $\frac{-8-\alpha}{-8-\beta}$ however I do not know how to proceed further. I might be doing this completely wrong and my apologies for that.
Another solution came to me that if $\alpha$ and $\beta$ are roots of the other unknown equation. I can somehow manipulate that to find the answer. But I don't think that will work. All help is appreciated thank you.
 [/INST] If $α$ and $β$ are the roots of the quadratic equation $ax^2+bx+c=0$ ,that is, $x^2+\frac{b}{a}x+\frac{c}{a}=0$ ($a$ is not equal to zero), then the equation can be written as $x^2-(α+β)x+αβ=0$.     
The equation whose roots are $\frac{α}{β}$ and $\frac{β}{α}$ is: $x^2-(\frac{α}{β}+\frac{β}{α})x+(\frac{α}{β})(\frac{β}{α})=0$.
$\implies x^2-(\frac{α^2+b^2}{αβ})x+1=0$
$\implies x^2-(\frac{(α+b)^2-2αb}{αβ})x+1=0$
$α$ and $β$ are the roots of the equation $x² + 8x - 5 = 0$
$α+β=-8$
$αβ=-5$
Therefore the required equation is $x^2-(\frac{(-8)^2-2(-5)}{-5})x+1=0$
$\implies x^2-(\frac{74}{-5})x+1=0$
$\implies x^2+\frac{74}{5}x+1=0$
$\implies 5x^2+74x+5=0$
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