Question

How do you factor `m^2-10my+25y^2`?

Answer 1

`(m-5y)(m-5y) = m^2-10my+25y^2`

Explanation:

One of our first clues in factoring this expression is that the first and last terms are perfect squares:

`sqrt(m^2) = m`

and

`sqrt(25y^2) = 5y`

In the most general case, we are looking for a solution in the form of

`(am+by)(cm+dy) = m^2-10my+25y^2`
`acm^2 + (bc+ad)my + bdy^2 = m^2-10my+25y^2`

From the first term we can see that `a*c = 1`. Assuming `a` and `c` are integers, they must be both either `+1` or `-1`. Let's make them `+1` for now and continue (it actually doesn't matter which we choose at this point - can you see why?):

`m^2 + (b+d)my + bdy^2 = m^2-10my+25y^2`

From the last term we see that `b*d = 25`. From the second term, we must also have `(b+d) = -10`. The obvious solution for this is to have `b=d=-5`. Therefore we have the solution

`(m-5y)(m-5y) = m^2-10my+25y^2`

Once we have worked through this type of problem, we could have probably started by guessing this solution from the first two observations.