Question

Is it possible to factor `y=3x^2 + 9x-6`? If so, what are the factors?

Answer 1

`3x^2+9x-6 = 3(x+3/2-sqrt(17)/2)(x+3/2+sqrt(17)/2)`

Explanation:

Complete the square then use the difference of squares identity:

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

with `a=(2x+3)` and `b=sqrt(17)` as follows:

`3x^2+9x-6 = 3/4(4x^2+12x-8)`

`color(white)(3x^2+9x-6) = 3/4((2x)^2+2(2x)(3)+3^2-17)`

`color(white)(3x^2+9x-6) = 3/4((2x+3)^2-(sqrt(17))^2)`

`color(white)(3x^2+9x-6) = 3/4(2x+3-sqrt(17))(2x+3+sqrt(17))`

`color(white)(3x^2+9x-6) = 3(x+3/2-sqrt(17)/2)(x+3/2+sqrt(17)/2)`