(I assumed the second term was -12x and not just -12 as given)
To find the vertex form, you apply the method of:
"completing the square".
This involves adding the correct value to the quadratic expression to create a perfect square.
Recall: (x-5)^2 = x^2 color(tomato)(-10)xcolor(tomato)(+25)" "larr color(tomato)(((-10)/2)^2 = 25)
This relationship between color(tomato)(b and c) will always exist.
If the value of c is not the correct one, add on what you need. (Subtract it as well to keep the value of the expression the same)
y = x^2 color(tomato)(-12)x+34" "larr ((-12)/2)^2=36 !=34
Adding 2 will make the 36 that is needed.
y = x^2 color(tomato)(-12)x+34 color(blue)(+2-2)" "larr the value is the same
y = x^2 color(tomato)(-12)x+color(tomato)(36) color(blue)(-2)
y = (x-6)^2-2" "larr this is vertex form
The vertex is at (6,-2)" "larr note the signs
How do you get to it?
y = color(lime)(x^2) color(tomato)(-12)x+36 color(blue)(-2)
y = (color(lime)(x)color(tomato)(-6))^2color(blue)(-2)
color(lime)(x = sqrt(x^2)) and color(tomato)((-12)/2 = -6) " check" sqrt36 = 6
