# Why is completing the square useful?

##### 1 Answer

To simplify quadratic expressions so that they become solvable with square roots.

#### Explanation:

Completing the square is an example of a Tschirnhaus transformation - the use of a substitution (albeit implicitly) in order to reduce a polynomial equation to simpler form.

So given:

#ax^2+bx+c = 0" "# with#a != 0#

we could write:

#0 = 4a(ax^2+bx+c)#

#color(white)(0) = 4a^2x^2+4abx+4ac#

#color(white)(0) = (2ax)^2+2(2ax)b+b^2-(b^2-4ac)#

#color(white)(0) = (2ax+b)^2-(sqrt(b^2-4ac))^2#

#color(white)(0) = ((2ax+b)-sqrt(b^2-4ac))((2ax+b)+sqrt(b^2-4ac))#

#color(white)(0) = (2ax+b-sqrt(b^2-4ac))(2ax+b+sqrt(b^2-4ac))#

Hence:

#2ax = -b+-sqrt(b^2-4ac)#

So:

#x = (-b+-sqrt(b^2-4ac))/(2a)#

So having started with a quadratic equation in the form:

#ax^2+bx+c = 0#

we got it into a form

So long as we are happy calculating square roots, we can now solve any quadratic equation.

Completing the square is also useful for getting the equation of a circle, ellipse or other conic section into standard form.

For example, given:

#x^2+y^2-4x+6y-12 = 0#

completing the square we find:

#(x-2)^2+(y+3)^2 = 5^2#

allowing us to identify this equation as that of a circle with centre