Quadratic

You start reading AI. Then you realize- hold, go back to probability. You start probability; hit binomial, law of Central Theorum, the Gaussian. The Gaussian first look like a triangle and then like an expanding and shrinking wave. Wow, I know this wave… it looks exactly like the chart of a stock. The Bolinger Band and the standard deviation. Yey!! I have hit the gold…dreams of money…. millions…. The stock market is cracked. I am next Jim Simons. And then you end up staring at Quadratic. Again. The first non-linearity. Brrrr…….

You begin from anywhere and you end up at Quadratic equation. It’s omnipotent. It has terms, coefficient, constants etc etc. I could see it but can’t understand it. Why? Because there are various ways independent components of an equation can be derived.

Look at coefficient. What is a coefficient? It represents Rate of Change. How do you get rate of change. Simple. Differentiate it.

Sum of roots ( i.e., α + β) = -coefficient of x / coefficient of x² ;

Product of roots (i.e., αβ ) = constant term / coefficient of x².

To obtain the quadratic polynomial, insert the sum and product of the roots into the expression x² – (sum of roots)x + (product of roots)

The roots are important as they represent the change in phase like from positive to negative or from negative to positive. It also represents the point at which a ball thrown in the air drops or in stock market the stock price crosses the mean or in profit maximization.

Change of Phase

The way I look at a quadratic equation is a variable on which multiple forces are working. Think of the dynamic variable to be an aeroplane. Multiple forces are working on it. While engine is providing the thrust (say 2x² ) and wind flowing in opposite direction is resisting the speed (say-4x from the equation). When you add these forces, you can find the final speed.

Or think it like the dynamic variable itself is combination of multiple forces like 2x² + 3x + 5 may look like 2(a+b)² + 3x + 5. In this case the x itself is result of certain squared function and as we know (a+b)² = a² + 2ab + b² . Where is taking us. Its a window to calculus. Look at my blog Approaching zero (https://hemantkhandade.com/2024/08/21/calculus-approaching-zero/)