Question

How do you factor completely `x^2-2xy-15y^2`?

Answer 1

`(x-5y)(x+3y)`

Explanation:

`x^2-2xy-15y^2`
Looking at the given algebraic expression we recognize from the first two terms that to factor the expression we have to apply the property:
`color(blue)((x-y)^2=x^2- 2xy+y^2)`
But in the given expression we need the term `y^2` so we can add it and subtract so that as if `0` is added to the expression.
Let's add `y^2` then subtract it
`=x^2-2xy-15y^2+y^2-y^2`
`=x^2-2xy+y^2-15y^2-y^2`
`=(x-y)^2-16y^2`
`=(x-y)^2-(4y)^2`
Checking the last step reached it is the difference of two squares that says:
`color(blue)(a^2-b^2=(a-b)(a+b))`
where in our case:`a=(x-y)` and `b=4y`
Then,
`(x-y)^2-(4y)^2`
`=(x-y-4y)(x-y+4y)`
`=(x-5y)(x+3y)`