What is the quadratic equation and what does it do?
In: 287
So I want to preface this that you’re asking about a specific math thing when you don’t have a good grasp on math so this is gonna be a bit tricky for you to get. It’s like you’re asking about what makes a specific guitar solo good when you don’t know what notes are. It’s gonna be a bit hard to get.
But I’ll try anyway (and there’s no shame in not knowing something!)
We can use an equation to represent the world around us. For example, say that I have three apples. Then I get two more apples. I now have five apples. I can represent that like this:
3+2=5
That’s a basic equation and is what most people think of when it comes to math. But that’s not all we can do. Sometimes there is an unknown, and we need to solve for it.
3+2=?
Now there’s an unknown in our equation. We don’t know what that question mark is until we solve for it. But the question mark doesn’t have to be at the end!
3+?=5
Now we have a slightly more complicated situation, we can’t just add the numbers together. We need to figure out how to set up the numbers and move them around so we can get the unknown. But sometimes things are a bit more complicated.
Imagine that you wanted to see not just how many apples you have, when you get two apples, but how many apples you have if someone keeps giving you apples, 2 every second. You can’t just do that like you did the previous time, because how many apples you have depends on how many you had when the previous guy gave you apples. So you need to set it up like this:
3 +2(the number of times given apples)=?
This is three (how many apples we start with) plus 2 apples, times the number of times someone gave us apples. But this is kind of long to write. Let’s simplify this by using place holders. Let’s use “x” instead of that long bit in the prentheses, and y instead of a question mark.
3+2x=y
Swapping it around, y=2x+3
This is a basic algebraic equation this is a linear equation. It describes something that has ONE thing that changes. In this case, it’s apples over time. Anything you could describe as something per something else is this kind of equation. Miles per hour for example.
Now it gets more complicated if you want to measure something that changes how fast it’s changing. For example, if you’re looking at how far you’ve gone in a car. You don’t drive at the same speed the whole time, so you can’t use the same kind of formatting. I’m not going to get into the specifics, but an example of how this equation would work is:
y=6(x^2)+5x-2
Now those numbers are fine for this case, but what if the numbers are different? If we want to talk about this kind of format more generically, we need placeholders. We could just use # instead of the number, but what if we want to talk about the first number specifically? Let’s give each one a letter.
y=a(x^2)+bx+c
Now if we want to know what that unknown “x” is, we need to rearrange it into the quadratic equation. This lets us figure out what that number is if it would be really hard otherwise. Hopefully that was helpful!
The quadratic equation is an algorithm (method you can use to to solve) a certain type of problem that is somewhat common. If you can get your equation to look a certain way you can use the quadratic equation to solve for an unknown number, assuming you know all of the other numbers.
Care for a geometric explanation? The most important objects in geometry (or in problems for which you can make a geometric drawing) are straight lines and circles. Solving problems involves calculating where curves intersect.
Finding out where two straight lines intersect is a *linear* equation — that’s easy to solve.
Quadratic equations are how you find the intersections between a circles and a line, or between two circles.
It so happens that problems that boil down to that operation are very common.
A quadratic equation describes a curve on a graph. The most important thing that they do is allow you to concisely describe a curve rather than needing to describe every point on the curve.
One of the most common uses of quadratics is in compression of information. Polynomials (of which quadratics are the most basic) are used in the compression algorithms for digital music, for example.
Instead of recording every single sample at a given sample rate, the averages between samples are recorded as polynomials which take up much less space.
When you chuck something it follows a path called a parabola. Say you throw a ball, now it’s going 12 meters per second up and 15 meters per second sideways and you throw it from 2 meters off the ground and you want to calculate where it will land. Say gravity is -10 m/s^2, that is: every second in the air the ball accelerates down with 10 more meters per second of downward speed. That’s all gravity is.
Now I’m going to start running through some algebra, you don’t need to understand every step but I want you to understand where the letters come from to understand why the quadratic formula has so many letters in it.
First we need to write the ball’s position as algebra.
Horizontally the ball just keeps going sideways at 15 m/s. Gravity can’t change horizontal speed.
So x is the ball’s position horizontally and t is seconds. x=15t is how we say its horizontal position is just 15 * the number of seconds since you threw it
Now to find the balls height over time. So y is the ball’s height and t is seconds: y = -10*t*t + 12*t + 2
Which is how you write that the ball is accelerating down at 10m/s^2 and starts out at 12 m/s of speed going up and starts 2 meters off the ground.
But we want to know *where* the ball lands not *when* so we need to get rid of seconds and just work in meters. Algebra helps us do that. Algebra tells us that if x=15t then t=x/15
And we use this to replace t with x/15. That means we’re replacing time with a formula for horizontal distance
y = -10*t*t + 12*t + 2 becomes y = -10*x/15*x/15 + 12*x/15 + 2
If you know some more algebra that simplifies to y = -(2/45)*x^2 + (4/5)*x + 2
Now we have an equation that shows the path of the ball. We can put in x (horizontal distance) and get y (vertical height) and you can graph this and see a real path of the ball.
Now we want to find where the ball lands. You could guess and check. Try out values of x and see how close you can get y to be 0 but we have an equation: the quadratic formula that solves it for us.
You need an equation in the form y=a*x^2 + b*x + c and it will tell you where it crosses y=0. For us that means where it hits the ground.
Our equation is in that form so a=-2/45, b=4/5, and c=2
We plug it all into that big ugly equation and we get two answers?? One says the ball lands at 20.225 meters after we threw it and the other answer says it lands -2.225 meters behind us.
There is a plus or minus in the quadratic equation because parabolas cross y=0 twice. In this case the full parabola crossed y=0 before we even threw it.
Parabolas can also just not cross y=0 at all, like if you threw a ball from under zero and the throw just didn’t make it up to the roof where zero is. In this case the quadratic formula will give you two square roots of negative numbers or two “imaginary” numbers. For now you can throw these out and just say it never crosses y = 0
The quadratic equation is a specific way to define a relationship between two values.
In the case of the “quadratic”, the relationship specifically has to do with the one value being related to the “square” of the other value – such as “9” is “3” squared.
That and a little bit more – it is related to the square of the if value, plus the value, plus some fudge factor.
There are other specific types of relation ships as well
The blue car was going twice as fast as the red car – this is a “linear” relation
The area of a circle is proportional to the square of its radius – this is a “quadratic” relation. Allow me to explain.
Area = (pi) * r^2 + (0)*r + 0
Here, your “a” is pi, “b” is 0 and “c” is 0
There are other types of relations as well, such as “exponential”. This can be like – the number “10,000” is the the number “10” times itself many times! (In the event that it is related to the other number times itself once, we call it “quadratic”. )
Here are some relation examples as an example
50 = 5*x is linear (x is 10)
200 = 2*x^2 is quadratic (x is 10).
10,000,000,000 = 10^x is exponential (x is 10)
NOTE: a quadratic equation can have linear” component (b*x), and a constant (c) it doesn’t have to.
This will not help you at all, but when I read your question, I heard it in my head in an Arnold Schwarzenegger voice, saying
“Who is quadratty, and what does he do?”
One of my goals in life is to actually recognize a real world application for the quadratic equation and use it organically
As mentioned by some others, the quadratic equation is just the formula for finding x when an equation looks like:
ax^2 + bx + c = 0
The process of finding this formula is straightforward and anyone who has just completed an Algebra class in Middle or High school should be able to do it.
However, in addition to being a useful formula for finding x in its own right, it also has a useful property of giving us information about the solution without having to fully solve it.
In the formula (which is repeated in many other answers), the part under the square root sign is:
b^2 – 4ac
Just solving this part by itself tells us about the values of x we will find if we fully solve for x.
This value by itself is called the determinant. If it is positive, there are 2 real values for x. If it is 0, there is only 1 real value for x. If it is negative, then there are no real values for x. What the last part means is that there is no real number value for x that will satisfy the equation.
This means only part of the formula needs to be solved to know if there is actually a possible real number solution.
Something a little more deep to mention, since others have done good explanations of what the equations does:
A lot of processes in life follow functions. That just means there is a way to mathematically write what they do as we play time forward. But we often don’t know the function they follow, or know it but it’s really complicated.
So we make a different function that is very close, but a lot simpler. Like smoothing a line a bit so you can draw it with a ruler. One of the ways we do that is called “Taylor approximation”.
How that works doesn’t matter, but a Taylor approximation is a pretty good and pretty natural way to get a function that is good enough. Now why is this relevant?
Say you have some function y=f(x), where we have no idea what f(x) is. A mathematician will do a bit of work and be able to get a number a.
y=a is our “zeroth” approximation. It’s very bad because we are essentially assuming nothing changes.
So we do a bit more work and get two numbers: a and b.
y=ax+b is our “first” approximation. It’s better, but still not very good, since it’s a straight line and not much in nature is straight.
So we do a bit more work and get three numbers: a, b, and c.
y=ax^(2)+bx+c is our “second” approximation, and already pretty good in a small area. It has a curve, a slope, and a height. but we can do better.
And so on…
The reason we use the quadratic formula so much is that in a lot of situations the second Taylor approximation is a very good tradeoff between accuracy and complexity. That means we need a lot of quadratic formulas to solve our smoothed out functions.
(Part of the reason we often stop at the second Taylor approximation is that formulas for solving more than quadratics are **really** hard and often impossible. So it’s all related.)
Latest Answers