Question

How do you factor `x^2+9x+3`?

Answer 1

`x^2+9x+3 = (x+9/2-sqrt(69)/2)(x+9/2+sqrt(69)/2)`

Explanation:

Complete the square then use the difference of squares identity:

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

with `a=(x+9/2)` and `b=sqrt(69)/2` as follows:

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

`color(white)(x^2+9x+3) = (x+9/2)^2-81/4+12/4`

`color(white)(x^2+9x+3) = (x+9/2)^2-69/4`

`color(white)(x^2+9x+3) = (x+9/2)^2-(sqrt(69)/2)^2`

`color(white)(x^2+9x+3) = ((x+9/2)-sqrt(69)/2)((x+9/2)+sqrt(69)/2)`

`color(white)(x^2+9x+3) = (x+9/2-sqrt(69)/2)(x+9/2+sqrt(69)/2)`