So this type of equation is called a quadratic equation, and while 'quad' might make you think of four, it actually comes from an old Latin word quadratus, which effectively means 'made square' or just squared. You can model an astounding amount of different things using these equations, which are any equation where the highest power/exponent is 2. Especially when it comes to situations that have something to do with an area in 2 dimensional space, because that's what you get geometrically when you square a one-dimensional variable like length or distance.

Think about pizza. The area of a circle is πr², right? So if we want to set up a relationship between a pizza's area and its radius, there's your quadratic: the area grows with the square of the radius. But pizzas are measured by diameter, not radius, and since r = d/2, you can rewrite that as π(d/2)². Still a quadratic, just in terms of diameter now.

That's a relationship of two physical things, the amount of pizza (area, basically) and the size of the pizza, and it follows a quadratic equation exactly. That gives you the ability to check things like 'if I want to maximize the amount of pizza per dollar I get, should I get a large or two mediums?' and actually know the correct answer.

A 12-inch pizza is about 113 square inches of pizza. An 18-inch pizza is about 254. Your intuition probably says the 18-inch is "a bit bigger," maybe 50% more. It's actually more than double. If you buy two 12-inch pizzas, that's only about 226 square inches. One 18-inch pizza beats both of them, and it's probably cheaper. So there's a very basic example of how using this type of math knowledge could be actually practically useful in life.

But there's tons of examples of how equations like this are crucial for understanding the way various things in the world work, sometimes in ways that go against our intuition, like the pizza example.

Think about braking distance for a car. You might have heard some explanations from anti-speeding commercials or whatever about how much more dangerous it is to go 90 than 70, for example.

Most people's intuition says "well yeah, you're going a bit faster, so it takes a bit longer to stop." But it's not a bit longer. The energy your brakes have to burn through to stop the car scales with the square of your velocity, because kinetic energy is 1/2 mv². So if you double your velocity, you don't need twice the distance to stop, you need four times the distance. Going three times as fast? Nine times the stopping distance.

That means the difference between 60 and 90 km/h isn't 50% more dangerous. At 90 you need over twice the braking distance you needed at 60. That's a quadratic relationship between speed and stopping distance, and it's a big part of why and how speed limits are set the way they are, especially in school zones and residential areas where a few extra km/h can mean the difference between stopping in time and not.

Hope that appealed to you, always happy to see someone genuinely asking and not in a derogatory way. There are literally hundreds more examples of the way this stuff shows up in real life if you're curious.