Question

How do you know if `2x^2 + 7x + 5` is factorable?

Answer 1

You have to try to solve the associated equation:

`ax^2+bx+c=0`

and if the `Delta=b^2-4ac` is positive it could be factored:

`a(x-x_1)(x-x_2)` where `x_1 and x_2` are the two solution:

`x_(1,2)=(-b+-sqrtDelta)/(2a)`.

If `Delta` is zero, than it it a square of a binomial, and if `Delta` is negative, it couldn't be factored.

So, in our case:

`Delta=7^2-4*2*5=49-40=9`, so we can find the two solutions:

`x_(1,2)=(-7+-3)/4`

and so:

`x_1=(-7-3)/4=-10/2=-5/2`

`x_2=(-7+3)/4=-4/4=-1`

it could be factored:

`2(x+5/2)(x+1)` or `(2x+5)(x+1)`.