The discussion revolves around the algebraic expression $$ 4a^4 + 8 a^3 + 8 a^2 + 4a + 1 $$ and its equivalence to the perfect square $$ ( 2a (a+1) + 1 )^2 $$. Participants are attempting to understand the derivation of this identity.
There are various approaches being discussed, including factoring the LHS and rewriting terms to identify a perfect square. Some participants have suggested specific algebraic manipulations, but no consensus has been reached on a definitive method.
Participants have noted the absence of a formal homework statement or relevant equations, which may impact the clarity of the problem context.
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?