Can someone please give me a practical use of completing the square?!

Hmm.. I assume you mean writing a² x² + bx + c

As (ax + (b/2a))² - (b/2a)² + c

Well... you can look at c - (b/2a)² . As the first term is always positive, then if this is less than zero, you know no roots exist. If it is zero, the roots are the same and if it is more than one, then two roots exist (a root being where the equation is zero for a certain value of x)

Although you probably haven't done it yet, it can also be useful in very complicated integrals, by using it you can integrate e^{a quadratic equation} quite easily. Without this, it is VERY hard...

Hmm.. It can make the graph easier to draw and finally, who says maths has to be useful... If we are honest, most of it is just there for the fun of it!!!

Bye for now, Chris. If anyone has any other uses, feel free to add them!

How do you integrate e^{quadratic expression} ?

This is how I would do it:

First, let the quadratic expression be ax² +bx+c.

Then, completing the square to give the exponent in the form au² +d, where u=x+(b/2a).

If a is negative, the answer would be an error function, in fact, I found that it is

e^d Erf([x+b/2a]sqrt[|a|])/sqrt(|a|)+constant

but if a is positive, hm... I don't know, perhaps expressing the integrand as power series and integrate.