The algebraic equation $$ 4a^4 + 8 a^3 + 8 a^2 + 4a + 1 = ( 2a (a+1) + 1 )^2 $$ represents a perfect square. Participants in the discussion emphasized the importance of manipulating the left-hand side (LHS) to reveal its structure as a square. Key strategies included factoring the LHS and rewriting terms, such as expressing $$8a^2$$ as $$4a^2 + 4a^2$$. Ultimately, the right-hand side (RHS) must conform to the form $$ (\alpha a^2 + \beta a + 1)^2 $$ to derive the necessary coefficients.
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Well, what happens when you multiply out the RHS? Can you show those steps to see how close you get to the LHS?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.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##.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?