What is the quadratic equation and what does it do?
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Did anyone else read the title like Arnold Schwarzenegger says “who is your daddy and what does he do”?
A quadratic equation describes a parable, a line that moves x squares in one direction and x² squares in a perpendicular direction to that. Many natural phenomena can be described with that kind of curve, falling objects are the most obvious example because both their speed (and, therefore the friction they experience) grows as the objects fall.
A quadratic equation is just a specific type of polynomial equation; one that looks like ax^2 + bx + c = 0, if solving for particular values (called roots). In general, a quadratic just refers to the expression on the left hand side of the equation.
Quadratics are useful for many things. I’d say the two “common” ways they come up in applied situations is, firstly, in problems involving the area of something with unknown side lengths that are related to each other, and secondly in problems involving something analogous to constant acceleration (often it literally is constant acceleration, such as gravity to a good approximation).
In areas outside of applied mathematics, they are everywhere for many many other reasons. In discrete situations, you have the idea of quadratic residues and quadratic non-residues, which for example come up in searching for prime numbers. This furthermore relates to ways that primes and number theory is used in cryptography.
To give an ELI15:
Basic algebra equations are usually something like y = 2x + 4, which means that the variable y is two times x plus four.
So if y is how many pancakes you get from x number of eggs, then if you have 3 eggs, you’ll get 10 pancakes.
The question is, if you have a certain amount of pancakes, can we figure out how many eggs it took?
In this case what we do is the steps from before, but now backwards and opposite. The original was y is two times x plus four. The opposite of plus is minus, and times is divided. So backwards and opposite, it’s y minus four divided by two equals x.
As an equation: (y-4)/2 = x. So if we have 12 pancakes, we take away 4, we have 8 now. Then we divide 8 by 2, giving us 4. We used 4 eggs to get 12 pancakes.
Quadratic equations are ones that look like y = 4x^2 + 2x + 1. The quadratic formula is the same thing as the backwards and opposite step in basic algebra equations, except only when we have 0 as our y.
There are a bunch of equations where it’s important to find the x value for 0 = ax^2 + bx + c, which is why the quadratic formula finding zero is very common.
Maybe an oversimplification but ELI5 hopefully. Think of throwing a ball up into the air. Starts heading up eventually stops at the top, and starts heading down until it hits the ground and stops again.
Quadratic would help answer questions like:
When (as a time factor) will the ball be 2 meters above the ground? Well at one point when it’s heading up it will hit 2 meters, and another when it’s heading down, but at different times.
Or, when will the ball have 0 speed? Realistically at the top of the arc and when it hits the ground.
Or, when will the ball be travelling 0.5 meters per second, once as it’s slowing down but heading up, and once when it’s speeding up and heading down.
Rinse and repeat for parabolic activities while solving for singular points.
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