Question 123071

{{{y=x^2+4 x+9}}} Start with the given equation



{{{y-9=x^2+4 x}}}  Subtract {{{9}}} from both sides



Take half of the x coefficient {{{4}}} to get {{{2}}} (ie {{{(1/2)(4)=2}}}).


Now square {{{2}}} to get {{{4}}} (ie {{{(2)^2=(2)(2)=4}}})





{{{y-9=x^2+4x+4-4}}} Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of {{{4}}} does not change the equation




{{{y-9=(x+2)^2-4}}} Now factor {{{x^2+4x+4}}} to get {{{(x+2)^2}}}



{{{y=(x+2)^2-4+9}}} Now add {{{9}}} to both sides to isolate y



{{{y=(x+2)^2+5}}} Combine like terms



So what we've done is complete the square and convert {{{y=x^2+4 x+9}}} into 
{{{y=(x+2)^2+5}}}



So this means that {{{x^2+4x+9=(x+2)^2+5}}} which in turn also means {{{(x+2)^2+5=(x+a)^2+b}}}



Here we can see that {{{a=2}}} and {{{b=5}}}



Answer:



So our answers are: {{{a=2}}} and {{{b=5}}}