Question

How do you factor `y=x^2 + 35x + 36` ?

Answer 1

You can not factor that trinomial, unfortunately.

Explanation:

You can always see if a quadratic function like the one above is factorable by using the discriminant of the quadratic formula:

`b^2` - 4ac
`35^2` - 4(1)(36)
1120 - 144
976

Since the final result, 976, is not a perfect square, this trinomial cannot be factored into even factors.

Hopefully you understand now.

Answer 2

This quadratic can only be factored using irrational coefficients:

`x^2+35x+36`

`=(x+35/2-sqrt(1081)/2)(x+35/2+sqrt(1081)/2)`

Explanation:

The difference of squares identity can be written:

`a^2-b^2 = (a-b)(a+b)`

So:

`x^2+35x+36`

`=x^2+35x+(35/2)^2-(35/2)^2+36`

`=(x+35/2)^2-1225/4+144/4`

`=(x+35/2)^2-1081/4`

`=(x+35/2)^2-(sqrt(1081)/2)^2`

`=((x+35/2)-sqrt(1081)/2)((x+35/2)+sqrt(1081)/2)`

`=(x+35/2-sqrt(1081)/2)(x+35/2+sqrt(1081)/2)`

We cannot simplify `sqrt(1081)` further since `1081=23*47` has no square factors.