### how to use a parabola for multiplication

It's possible to multiply any two real numbers using just a parabola and some straight lines. Pick two numbers, *x*_{1} and *x*_{2} say, extend a line vertically until you hit the parabola, join both of these points and where this line intersects the y-axis, is the product *-ax*_{1}x_{2}, where *a* is the coefficient of the *x*^{2} term in the parabola.

In the example below, I choose *x*_{1} = 7 and *x*_{2} = -3, and a parabola described by *y = x*^{2}.

As you can see, the line joining points (-3,9) and (7,49) intersects the y-axis at (0,21), which is equal to -(1)(7)(3).

The choice of parabola is irrelevant here, as long as the constant term is 0. I will use the general from of a parabola with no constant term, that is *y = ax*^{2}+bx, in the proof below to demonstrate this.

**Proof:**

Claim: The y-intersect of the line joining two points on a parabola described by *y = ax*^{2}+bx is the negative of the product of the *x* components of the same two points on the parabola and the coefficient of the *x*^{2} term describing the parabola.

Let *x*_{1} and *x*_{2} be two arbitrary points on the x-axis. Then, both

*y*_{1} = ax_{1}^{2}+bx_{1}

*y*_{2} = ax_{2}^{2}+bx_{2}

are points lying on the parabola, and the equation of the line joining them is given by *l = mx+c*, where *m* is the slope and is described by:

*m = (1/x*_{1}-x_{2})(ax_{1}^{2}+bx_{1} - (ax_{2}^{2}+bx_{2}))

m = (1/x_{1}-x_{2})(ax_{1}^{2}+bx_{1} - ax_{2}^{2}-bx_{2})

m = (1/x_{1}-x_{2})(a(x_{1}^{2}-x_{2}^{2})+b(x_{1}-x_{2}))

m = (1/x_{1}-x_{2})(a(x_{1}-x_{2})(x_{1}+x_{2})+b(x_{1}-x_{2})) (Difference of squares)

m = a(x_{1}+x_{2})+b

Therefore, the line *l* joining *y*_{1} and *y*_{2} on the parabola is given by *l = (a(x*_{1}+x_{2})+b)x+c. It just remains to calculate the value of *c*, which is the y-intersect of *l*.

Since we already know that (*x*_{1},ax_{1}^{2}+bx_{1}) is a point on the parabola, filling these values into the equation of *l*, we get:

*ax*_{1}^{2}+bx_{1} = (a(x_{1}+x_{2})+b)x_{1}+c

ax_{1}^{2}+bx_{1} = ax_{1}^{2}+ax_{1}x_{2}+bx_{1}+c

0 = ax_{1}x_{2}+c

c = -ax_{1}x_{2}

**Note:**

The parabola can't have a non-zero constant term, as this value would just shift the parabola up or down vertically, which would only affect the value of the y interesection. For more on parabolas, Matt Parker has a video on his youtube channel: There is only One True Parabola