Question 156156
{{{x^2+6x+16=(x+a)^2+b}}} Start with the given equation



{{{x^2+6x+16=x^2+2ax+a^2+b}}} FOIL



{{{6x+16=2ax+a^2+b}}} Subtract {{{x^2}}} from both sides



Notice how the term {{{2ax}}} is the <b>only</b> term on the right side that has an "x" in it. So this means that {{{6x=2ax}}}


{{{6x=2ax}}} Start with the given equation



{{{3=a}}} Divide both sides by {{{2x}}} to isolate "a"



So the first answer is {{{a=3}}}


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Looking back at {{{6x+16=2ax+a^2+b}}}, the terms {{{a^2+b}}} do not have an "x" term in them at all. So this means that {{{16=a^2+b}}}



{{{16=a^2+b}}} Start with the given equation



{{{16=3^2+b}}} Plug in a=3



{{{16=9+b}}} Square 3 to get 9



{{{7=b}}} Subtract 9 from both sides



So the second answer is {{{b=7}}}



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Answer:


So the solutions are {{{a=3}}} and {{{b=7}}}





Check:


{{{x^2+6x+16=(x+a)^2+b}}}



{{{ x^2+6x+16=(x+3)^2+7}}}



{{{ x^2+6x+16=x^2+6x+9+7}}}



{{{ x^2+6x+16=x^2+6x+16}}}


{{{0=0}}} works