Quick questions on Geometry using polynomial techniques: tangency as a double root of P(x) minus Q(x), chord midpoints from the sum of roots, tangents from an external point, a line through a fixed point on a cubic meeting it twice more, and a common tangent to two curves, all without calculus

Take any two curves $y = P(x)$ and $y = Q(x)$. They meet exactly where $P(x) = Q(x)$, that is where the single equation

From a point outside a parabola there are exactly two tangent lines, and the double-root condition finds both at once. Write the general line through the external point with unknown gradient $m$, form the quadratic intersection equation, and set its discriminant to zero. The discriminant condition is itself a quadratic in $m$, and its two solutions are the gradients of the two tangents. (A point inside the parabola gives a negative discriminant in $m$, hence no real tangents; a point on the parabola gives a repeated $m$, the single tangent there.)

A favourite Extension 1 set-up: a point $P$ lies on a cubic, and a line of variable gradient $m$ through $P$ meets the cubic at two further points $A$ and $B$. Because $P$ is on both the line and the curve, its x-coordinate is automatically a root of the cubic intersection equation, for every $m$. That leaves a clean structure: the three roots are (x-coordinate of $P$), $\alpha$, $\beta$, and the sum of all three is fixed by the coefficient of $x^2$. So $\alpha + \beta$, and therefore the midpoint of $AB$, is pinned down regardless of $m$.