Given 2x^2+8x
Extract the coefficient of x^2 as a factor
color(white)("XXXX")=(2)(x^2+4x)
Half of the coefficient of x is 1/2xx4 = 2
So the square of half the coefficient of x is 2^2 = 4
Add (and subtract) the square of half the coefficient of x
color(white)("XXXX")= (2)(x^2+4x+2^2 -4)
Which could be written as
color(white)("XXXX")=(2)((x+2)^2 -4)
In case you were wondering "why the square of half the coefficient of x?)
color(white)("XXXX")We are trying for a squared binomial of the form (x+a)^2
color(white)("XXXX")Since
color(white)("XXXX")color(white)("XXXX")(x+a)^2 = x^2+2ax+a^2
color(white)("XXXX")Given the first 2 terms in the form:
color(white)("XXXX")color(white)("XXXX")x^2+d
color(white)("XXXX")color(white)("XXXX")color(white)("XXXX")(d=2a)
color(white)("XXXX")We need to add (d/2)^2 to get a "squared form"
