The roots of the equation $x^2 - 6x + 7 = 0$ are $α$ and $β$. Find the equation with roots $α + 1/β$ and $β + 1/α$. Quadratic equation question, as specified in the title.

The roots of the equation $x^2 - 6x + 7 = 0$ are $α$ and $β$. Find the equation with roots $α + \frac{1}{β}$ and $β + \frac{1}{α}$.

I gather that $α + β = -\frac{b}{a} = \frac{6}{1} = 6$ and that $αβ = \frac{c}{a} = \frac{7}{1} = 7$. Do I need to convert $α + \frac{1}{β}$ and $β + \frac{1}{α}$ into a format whereby I can sub in the values for adding together or multiplying $α$ and $β$ ? If so, how ?

$(x-\alpha)(x-\beta) = x^2 - (\alpha + \beta)x + \alpha\beta=0\\
(\alpha + \beta) = 6\\
\alpha\beta = 7$
$(x-\alpha - \frac 1\beta)(x-\beta - \frac 1\alpha) = x^2 - (\alpha + \beta + \frac 1\alpha + \frac 1\beta)x + (\alpha\beta + 2+ \frac {1}{\alpha\beta})=0\\
x^2 - (\alpha + \beta + \frac{(\alpha + \beta)}{\alpha\beta})x + (\alpha\beta + 2+ \frac {1}{\alpha\beta})=0\\
x^2 - (6 + \frac 67)x + (9 + \frac 17)=0$
