Question

please guide me through COMPLETING THE SQUARE

Answer 1

\[a^{4} + 2a ^{2}b ^{2} + 9b ^{4}\]

Answer 2

if we ignore the \(9b^4\) term and make
\[a^2+2a^2b^2\] in to a perfect square we would add \(b^4\) because
\[a^4+2a^2b^2+b^4\] is a perfect square, namely
\[(a^2+b^2)^2\]

Answer 3

so one way to write this is
\[(a^2+b^2)^2+8b^4\] but i am not sure that is what you are looking for

Answer 4

hii, we just have to factor it completely :) that's all

Answer 5

oh ok we can do that too
it is going to require a trick called sophie germain

Answer 6

the gimmick is to note that
\[\large (a^2+3b^2)^2=a^4+6a^2b^2+9b^4\]

Answer 7

then write
\[a^4+2a^2b^2+9a^4\] as
\[a^4+6a^2b^2+9b^4-4a^2b^2\]

Answer 8

i.e add and subtract \(4a^2b^2\)
with me so far?

Answer 9

yes yes :)

Answer 10

ok now we know how to factor the first part, as it is a "complete square" so we have
\[(a^2+3b^2)^2-4a^2b^2\]

Answer 11

can I present my own solution too? to see if it's the same as yours? :)

Answer 12

and now you have the difference of two squares

Answer 13

sure

Answer 14

yeah (a-b) and (a+b) ? :)

Answer 15

YES