SOLUTION: Find numbers a and b so that x^2+4x+9=(x+a)^2+b

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Question 123071: Find numbers a and b so that x^2+4x+9=(x+a)^2+b
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

y=x%5E2%2B4+x%2B9 Start with the given equation


y-9=x%5E2%2B4+x Subtract 9 from both sides


Take half of the x coefficient 4 to get 2 (ie %281%2F2%29%284%29=2).

Now square 2 to get 4 (ie %282%29%5E2=%282%29%282%29=4)




y-9=x%5E2%2B4x%2B4-4 Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of 4 does not change the equation



y-9=%28x%2B2%29%5E2-4 Now factor x%5E2%2B4x%2B4 to get %28x%2B2%29%5E2


y=%28x%2B2%29%5E2-4%2B9 Now add 9 to both sides to isolate y


y=%28x%2B2%29%5E2%2B5 Combine like terms


So what we've done is complete the square and convert y=x%5E2%2B4+x%2B9 into
y=%28x%2B2%29%5E2%2B5


So this means that x%5E2%2B4x%2B9=%28x%2B2%29%5E2%2B5 which in turn also means %28x%2B2%29%5E2%2B5=%28x%2Ba%29%5E2%2Bb


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


Answer:


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