Question

How do you factor the expression `x^2+3x+1`?

Answer 1

`x^2+3x+1 = (x+3/2-sqrt(5)/2)(x+3/2+sqrt(5)/2)`

Explanation:

We can complete the square and use the difference of squares identity, which may be written:

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

with `a=(x+3/2)` and `b=sqrt(5)/2` as follows:

`x^2+3x+1 = (x+3/2)^2-9/4+1`

`color(white)(x^2+3x+1) = (x+3/2)^2-(sqrt(5)/2)^2`

`color(white)(x^2+3x+1) = ((x+3/2)-sqrt(5)/2)((x+3/2)+sqrt(5)/2)`

`color(white)(x^2+3x+1) = (x+3/2-sqrt(5)/2)(x+3/2+sqrt(5)/2)`