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)
