SOLUTION: if x^2+6x+16=(x+a)^2+b for all values of x, find the values of a and b

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Question 156156: if x^2+6x+16=(x+a)^2+b for all values of x, find the values of a and b
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
x%5E2%2B6x%2B16=%28x%2Ba%29%5E2%2Bb Start with the given equation


x%5E2%2B6x%2B16=x%5E2%2B2ax%2Ba%5E2%2Bb FOIL


6x%2B16=2ax%2Ba%5E2%2Bb Subtract x%5E2 from both sides


Notice how the term 2ax is the only 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%2B16=2ax%2Ba%5E2%2Bb, the terms a%5E2%2Bb do not have an "x" term in them at all. So this means that 16=a%5E2%2Bb


16=a%5E2%2Bb Start with the given equation


16=3%5E2%2Bb Plug in a=3


16=9%2Bb 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%5E2%2B6x%2B16=%28x%2Ba%29%5E2%2Bb


+x%5E2%2B6x%2B16=%28x%2B3%29%5E2%2B7


+x%5E2%2B6x%2B16=x%5E2%2B6x%2B9%2B7


+x%5E2%2B6x%2B16=x%5E2%2B6x%2B16

0=0 works