<s>[INST] Coordinate Geometry Parabola and Line Intersection Can someone please explain why $\alpha + \beta = -\frac{b}{a} = \frac{4}{3}$ for the image below?
Where $a$ and $b$ are constants that come from: $y^2-3y=5(\frac{1-y}{3})$

 [/INST] If $\alpha$ and $\beta$ are roots of the quadratic polynomial $ay^2+by+c$, then 
$$ay^2+by+c=a(y-\alpha)(y-\beta)$$
So
\begin{align}
ay^2+by+c&=ay^2-a(\alpha+\beta)y+a\alpha\beta
\end{align}
By equaling the coefficients of $y$ we get
\begin{align}
-a(\alpha+\beta)&=b\\
\alpha+\beta&=-\frac{b}{a}
\end{align}
</s>