When do you rearrange polynomials

You can rearrange polynomials through the complete the square form in order to easily transform graphs:

Where ax^2 + bx + c = 0

Becomes

a (x + b/2)^2 -ab^2/4 + c =0

Where a is the vertical stretch, the constant ( -ab^2/4 + c) the translation unites up or down and the b/2 being the translation horizontally.

You can also find turning points much easier using the complete the square form

Eg. (x-3)^2 + 5 = 0

Since any x value (negative or positive) will become positive

(x-3)^2 > or = 0

Therefore minimum point is where x = 3 to equate to 0 which would give a y value of 5 therefore minimum point is

(3,5)

Other reasons for rearrangement include setting the equation of the form y= ax^2 + bx +c to y=0 to find the roots (x-intercepts) of the equation.