Question

How do you factor `x^2 + 4xy - 5y^2`?

Answer 1

`(x-1y)(x+5y)`

Answer 2

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

Explanation:

All the information you need is in the expression.

Read from right to left.

Find the factors of 5 which subtract to give 4.
The signs will be different (because of the minus), there will be more positives (because of +)

5 is a prime number - the only factors are 1 x 5 and we see 5 -1 = 4.

We need +5 and -1 to give +4

This leads to the two brackets:

`(x" "y)(x" "y)" fill in the variables"`

`(x" "5y)(x" "1y)" fill in the factors"`

`(x + 5y)(x - y)" fill in the signs"`

Answer 3

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

Explanation:

`x^2+4xy-5y^2` is a homogeneous expression. We propose that can be formed by the product of two homogeneous expresions.
`x^2+4xy-5y^2 = (a x + b y)(c x + d y)`.

So

`x^2+4xy-5y^2 = a c x^2 + (bc+ad)xy +bd y^2`

so we have

#{ (1 = ac),(4=bc+ad),(-5=bd) :}#

We have three equations and four incognitas. Solving for `b,c,d` we obtain

`b = -a,c = 1/a,d = 5/a`

applying a feasible value for `a` as `a = 1` we obtain

`b =-1, c = 1,d = 5` so

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