Question

How do you solve `2x^3 + 6x^2 – x – 3 = 0`?

Answer 1

Factor by grouping and using the difference of squares identity to find:

`x=+-sqrt(2)/2` or `x=-3`

Explanation:

The difference of squares identity can be written:

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

We use this below with `a=sqrt(2)x` and `b=1`.

Factor by grouping then use difference of squares identity:

`0 = 2x^3+6x^2-x-3`

`=(2x^3+6x^2)-(x+3)`

`=2x^2(x+3)-1(x+3)`

`=(2x^2-1)(x+3)`

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

`=(sqrt(2)x-1)(sqrt(2)x+1)(x+3)`

So `x=+-1/sqrt(2) = +-sqrt(2)/2` or `x=-3`

graph{ 2x^3+6x^2-x-3 [-10.84, 9.16, -3.60, 7.2]}