Question

How do you factor `9k^2 + 30kn + 25n^2`?

Answer 1

`9k^2+30kn+25n^2 = (3k+5n)^2`

Explanation:

In general `(A* B )k^2+(A*Q+B*P)kn+(P*Q)n^2 = (Ak+Pn)(Bk+Qn)`
(This looks more complex than it is).

For the given expression:
`A*B = 9`
`color(white)("XXXX")`The (positive) factors of 9 are `(1,9) and (3,3)`
`color(white)("XXXX")`so we don't have too many possibilities to check.
`P*Q = 25`
`color(white)("XXXX")`The (positive) factors of 25 are `(1,25) and (5,5)`
`color(white)("XXXX")`again, there aren't a lot of possibilities.

The coefficient of the middle term
`color(white)("XXXX")` `30 = 3*5 + 3*5` (using only factors determined above).

So `A=3 and B=3`
and `P=5 and Q=5`

and the general factors `(Ak+Pn)(Bk+Qn)` become
`color(white)("XXXX")` `(3k+5n)(3k+5n)`