Question

How do you factor `4x^2 -20xy+25y^2`?

Answer 1

(2x-5y)(2x-5y).

Explanation:

`4x^2-20xy+25y^2`
`=4x^2-10xy-10xy+25y^2`
`=2x(2x-5y)-5y(2x-5y)`
`=(2x-5y)(2x-5y)`

Answer 2

`4x^2+20xy+25y^2=(2x+5y)^2`

Explanation:

Use the formula for the square of a binomial: `(a+b)^2=a^2+2ab+b^2`.

Both `4` and `25`, the coefficient of `x^2` and `y^2`, are perfect squares. This makes us think that the whole expression could be a perfect square: `4` is `2^2`, and `25` is `5^2`. So, our claim is that

`4x^2-20xy+25y^2` is `(2x-5y)^2`. Is it true? The only term to verify is `-20xy`, and it is indeed twice the product of `2x` and `-5y`. So, the conjecture was right.