How Did the Author Derive the Perfect Square from the Algebraic Equation?

AI Thread Summary
The discussion revolves around understanding how the expression 4a^4 + 8a^3 + 8a^2 + 4a + 1 can be factored into the form (2a(a+1) + 1)^2. Participants suggest starting from the right-hand side (RHS) and expanding it to see how it matches the left-hand side (LHS). They also recommend rewriting certain terms, like 8a^2, to facilitate finding a perfect square. The conversation emphasizes the need to manipulate the LHS to reveal its structure as a perfect square. Ultimately, the goal is to derive the perfect square form through algebraic manipulation.
MiddleEast
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Homework Statement
NA
Relevant Equations
NA
Hello,
While following problem solution found this $$ 4a^4 + 8 a^3 + 8 a^2 + 4a + 1 = ( 2a (a+1) + 1 )^2 $$
Trying to figure out how did author do it but failed.
Anyone?
 
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MiddleEast said:
Homework Statement: NA
Relevant Equations: NA

Hello,
While following problem solution found this $$ 4a^4 + 8 a^3 + 8 a^2 + 4a + 1 = ( 2a (a+1) + 1 )^2 $$
Trying to figure out how did author do it but failed.
Anyone?
Well, what happens when you multiply out the RHS? Can you show those steps to see how close you get to the LHS?
 
Thanks for quick reply. It is simple to start with R.H.S to L.H.S.
As per solution, they move from L.H.S to R.H.S, thats my question how to play with it?
Tried to get different common factors over and over, no success. It is supposed to move from L.H.S to R.H.S.
 
Just factor the LHS then. Do it in a first step to get <something> squared, and then look to simplify what is inside the squaring parenthesis...
 
MiddleEast said:
Homework Statement: NA
Relevant Equations: NA

Hello,
While following problem solution found this $$ 4a^4 + 8 a^3 + 8 a^2 + 4a + 1 = ( 2a (a+1) + 1 )^2 $$
Trying to figure out how did author do it but failed.
Anyone?
Try rewriting ##8a^2## as ##4a^2+4a^2##. I'm not sure what would motivate that other than trying to write the first three terms as a perfect square.
 
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MiddleEast said:
Homework Statement: NA
Relevant Equations: NA

Hello,
While following problem solution found this $$ 4a^4 + 8 a^3 + 8 a^2 + 4a + 1 = ( 2a (a+1) + 1 )^2 $$
Trying to figure out how did author do it but failed.
Anyone?
If the RHS is a perfect square, then it must be of the form ##(\alpha a^2 + \beta a +1)^2##.

Now, solve for ##\alpha, \beta##.
 
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