Given: y=-3x^2-5x+9
Write as: y=-3(x^2+5/3x)+9" ".................Equation(1)
Consider the (color(green)(x^2+5/3x)) part
We need to make this a ul("'perfect square'") but in 'forcing' it to do this we introduce a value that is not in the original equation. To correct this we have to turn it into 0 by subtraction or addition as appropriate by the same amount. Rather like a+2 being changed to (a+2) +3-3
color(green)(-3[x^2+5/3x] color(white)("ddd")->color(white)("ddd")-3[(x+5/(2xx3))^2]
color(green)(color(white)("dddddddddddddd")->color(white)("ddd")-3[x^2+5/3xcolor(red)(color(white)(.)ubrace(+(5/6)^2))])
color(white)("ddddddddddddddddddddddddddddddddd.d")color(red)(uarr)
color(white)("dddddddddddddddddddddddd")color(red)("The introduced error")
Substitute this into Equation(1)
color(green)(y=-3(x^2+5/3x)+9
color(white)("dddddddddddddddd")color(red)("The error")
color(white)("ddddddddddddddddd.d")color(red)(darr)
color(green)(y=ubrace(-3[x^2+5/3xcolor(red)(color(white)(.)+obrace((5/6)^2))])+color(blue)(k)+9)" " k is the correction
color(white)("ddddddddddd.d")color(green)(darr)
color(green)(y=color(white)("ddd")-3(x+5/6)^2color(white)("ddddd")+color(blue)(k)+9
The whole error is color(red)((-3)xx(5/6)^2)
color(green)(y=color(white)("ddd")-3(x+5/6)^2+color(blue)([3xx(5/6)^2]) +9)
color(white)()
y=-3(x+5/6)^2 +133/12
