Q.Prove that a straight line and parabola cannot intersect at more than two points.

A geometric question about how many points two curves can share often reduces, after substituting one curve's equation into the other, to a purely algebraic question: how many roots can this resulting polynomial equation have?

Worked illustration (line meets circle). With axes chosen so a circle is $x^2+y^2=r^2$ and a line is $y=mx+c$, substituting gives $(1+m^2)x^2+2mcx+(c^2-r^2)=0$ — a quadratic in $x$. A quadratic cannot have more than two roots, so a line and a circle meet in at most two points. Since the coefficients here are real, the quadratic's two roots are either both real (line crosses the circle at $2$ points, or is tangent with $1$ repeated point) or a non-real conjugate pair (line misses the circle entirely) — a line can never intersect a circle at exactly one simple, non-tangent point.

The identical substitute-then-count-roots idea proves further classical facts: …