Multiplication using Parabola

Theorem
Let $f \left({x}\right) = x^2$.

Let $A = \left({x_a, y_a}\right)$ and $B = \left({x_b, y_b}\right)$ be points on the curve $f \left({x}\right)$ so that $x_a \leq x_b$.

Then the line segment joining $A B$ will cross the $y$-axis at $-x_a x_b$.

Proof
Let $f \left({x}\right) = x^2$.

Then:


 * $f \left( {x_a} \right) = x_a^2$

and:


 * $f \left( {B_x} \right) = x_b^2$

Then the slope of the line segment joining $A B$ will be:

The result follows from taking the equation of the line defined by its slope and either point $A$ or $B$ to calculate the $y$-intercept.

Proceeding with point $A$: