What's the meaning of a quadratic equation?

I've studied it on high school but i've never understood it's meaning. So, what it means? I kinda "know" - this knowing is a little speculative - that it could be used to plot geometric figures and that some patterns of equations with higher degrees can be use to plot circles and some other geometric figures, can you add something to this or correct it? and, can you suggest something more?

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Have you tried searching Wikipedia. The following page en.wikipedia.org/wiki/Quadratic_equation gives a lot of information about the question what you are asking – user9413 May 3 '12 at 17:33
It doesn't have to mean anything, really... – J. M. May 3 '12 at 17:34
@Chandrasekhar I had it opened but it was not very clear to me. – Voyska May 3 '12 at 17:40
There is a difference between meaning and use. The equation "means" exactly what it says: this number times the square of the variable plus that number times the variable plus a third number equals zero. It can be used to model all sorts of things, the geometric aspect (parabolas) being only one. – Robert Israel May 3 '12 at 18:00
@Gustavo: Not necessarily. For example, trying to understand (a simplified version) of free fall leads to a quadratic equation; that's a physics problem, not a geometric one. – Arturo Magidin May 3 '12 at 18:32

A quadratic equation is a polynomial equation of degree 2. Quadratic equations are good for modelling parabolas and things which are naturally parabolic. One example of this is modelling the flight path of a projectile.

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A quadratic equation is a special type of equation. An equation is something like

$$x + 2 = 7$$

By solving this equation we mean that we want to find a number (or numbers) $x$ such that when I add $2$ to that number, then I get $7$. It is not hard to see that for this specific equation there is only one solution $x = 5$.

A quadratic equation is an equation of the form

$$ax^2 + bx + c = 0$$

where $a, b$, and $c$ are fixed numbers and $a \neq 0$. Again the question is: for which values of $x$ is this equation satisfied? So we simply want to find all the numbers $x$ such that when you evaluate/find $ax^2 + bx + c$ then you get $0$.

As an example, consider the quadratic equation

$$x^2 -1 = 0.$$

First note that this indeed is a quadratic equation by comparing it to the general form given above. We note that we get the equation by having $a =1, b = 0, c =-1$. We realize that there now is two distinct numbers $x$ such that $x^2 - 1 = 0$, namely $x =1$ or $x = -1$. There is a lot more to say about quadratic equations. One can for example find a general formula that gives you all the values that satisfy the equation. For more on this, you might (as suggested elsewhere) look over the Wikipedia article.

Now while there is an algebraic formula for solving a quadratic equations, there are also other methods. The Wikipedia article has a bit about a Geometric solution.

Edit: Here is a page about how to construct the solutions to a quadratic equation with ruler and compass.

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