Question 551125


{{{x^2+9x+12}}} Start with the given expression.



Take half of the {{{x}}} coefficient {{{9}}} to get {{{9/2}}}. In other words, {{{(1/2)(9)=9/2}}}.



Now square {{{9/2}}} to get {{{81/4}}}. In other words, {{{(9/2)^2=(9/2)(9/2)=81/4}}}



{{{x^2+9x+highlight(81/4-81/4)+12}}} Now add <font size=4><b>and</b></font> subtract {{{81/4}}}. Make sure to place this after the "x" term. Notice how {{{81/4-81/4=0}}}. So the expression is not changed.



{{{(x^2+9x+81/4)-81/4+12}}} Group the first three terms.



{{{(x+9/2)^2-81/4+12}}} Factor {{{x^2+9x+81/4}}} to get {{{(x+9/2)^2}}}.



{{{(x+9/2)^2-33/4}}} Combine like terms.



So after completing the square, {{{x^2+9x+12}}} transforms to {{{(x+9/2)^2-33/4}}}. So {{{x^2+9x+12=(x+9/2)^2-33/4}}}.



So {{{y=x^2+9x+12}}} is equivalent to {{{y=(x+9/2)^2-33/4}}}.