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Does a quadratic equation like $x^2 - ax + y = 0$ describe anything in the real world? (I want to know, if there is something in the same way that $x^2$ is describing a square.)

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Yes. Many many things. – Berci Jan 27 '13 at 21:51
Any reason why my tag-edit was reverted? This is not quadratic-forms. – mrf Jan 27 '13 at 21:58
Perhaps you can see these examples (there are many others): and… – Amzoti Jan 27 '13 at 22:12
@mrf: It got overwritten by George's suggested edit, which was probably meant for the previous version. I've reapplied your tag changes. – Rahul Jan 27 '13 at 22:12

The height $y(t)$ in meters of an object falling under gravity, after $t$ seconds, is given by $$y(t) = -4.9t^2 + v_0 t + y_0,$$ where $y_0$ is the object's initial height and $v_0$ is the object's initial vertical velocity. In particular, solving for $y(t)=0$ tells you when the object will hit the ground.

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Equation $x^2 - ax + y = 0$ represent for example ($a$ is real parameter) a families of parabolas. Parabola is trajectory of canon ball.

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Though not exactly same, depending upon value of a, following situations count as relevant. For deep explanation, see wikipedia.

  1. Bernoulli's Effect. This gives relation of velocity of fluid($u$), Pressure($P$), gravitational constant($g$) and height($h$), $$\frac{u^2}{g}+P=h$$

  2. Mandelbrot Set has Recursive Equation $$P_c=z^2+c, z\in\mathbb{R}$$ which is interesting and creates fractals which appear in nature.

  3. The descrete logistics equation is a quadratic equation which surprisingly generates chaos. This is the way population growth (be it bacteria or humans) is calculated. $$x_{n+1}=\mu x_n(1-x_n)$$

  4. Schrodinger's Equation

  5. Motion of Projectile as already mentioned

ed infinitum

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