Question 172409

{{{x^2-14x+50}}} Start with the right side of the equation.



Take half of the {{{x}}} coefficient {{{-14}}} to get {{{-7}}}. In other words, {{{(1/2)(-14)=-7}}}.



Now square {{{-7}}} to get {{{49}}}. In other words, {{{(-7)^2=(-7)(-7)=49}}}



{{{x^2-14x+highlight(49-49)+50}}} Now add <font size=4><b>and</b></font> subtract {{{49}}}. Make sure to place this after the "x" term. Notice how {{{49-49=0}}}. So the expression is not changed.



{{{(x^2-14x+49)-49+50}}} Group the first three terms.



{{{(x-7)^2-49+50}}} Factor {{{x^2-14x+49}}} to get {{{(x-7)^2}}}.



{{{(x-7)^2+1}}} Combine like terms.



So after completing the square, {{{x^2-14x+50}}} transforms to {{{(x-7)^2+1}}}. So {{{x^2-14x+50=(x-7)^2+1}}}.



So {{{y=x^2-14x+50}}} is equivalent to {{{y=(x-7)^2+1}}}.



So the equation {{{y=(x-7)^2+1}}} is now in vertex form {{{y=a(x-h)^2+k}}} where {{{a=1}}}, {{{h=7}}}, and {{{k=1}}}



Remember, the vertex of {{{y=a(x-h)^2+k}}} is (h,k).



So the vertex of {{{y=(x-7)^2+1}}} is (7,1).