To convert a quadratic from #y = ax^2 + bx + c# form to vertex form, #y = a(x - color(red)(h))^2+ color(blue)(k)#, you use the process of completing the square.
First, we must isolate the #x# terms:
#y - color(red)(81) = 4x^2 - 36x + 81 - color(red)(81)#
#y - 81 = 4x^2 - 36x#
We need a leading coefficient of #1# for completing the square, so factor out the current leading coefficient of 2.
#y - 81 = 4(x^2 - 9x)#
Next, we need to add the correct number to both sides of the equation to create a perfect square. However, because the number will be placed inside the parenthesis on the right side we must factor it by #4# on the left side of the equation. This is the coefficient we factored out in the previous step.
#y - 81 + (4 * ?) = 4(x^2 - 9x + ?)#
#y - 81 + (4 * 81/4) = 4(x^2 - 9x + 81/4)#
#y - 81 + 81 = 4(x^2 - 9x + 81/4)#
#y - 0 = 4(x^2 - 9x + 81/4)#
#y = 4(x^2 - 9x + 81/4)#
Then, we need to create the square on the right hand side of the equation:
#y = 4(x - 9/2)^2#
Because the #y# term is already isolated we can write this in precise form as:
#y = 4(x - color(red)(9/2))^2 + color(blue)(0)#
