Question

How do you factor `6x^2+7x-62`?

Answer 1

`f(x) = 6(x+7/12-sqrt(1537)/12)(x+7/12+sqrt(1537)/12)`

Explanation:

The difference of squares identity can be written:

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

We use this with `a=(12x+7)` and `b=sqrt(1537)` later.

Given:

`f(x) = 6x^2+7x-62`

`24f(x) = 144x^2+168x-1488`

`color(white)(24f(x)) = (12x+7)^2-49-1488`

`color(white)(24f(x)) = (12x+7)^2-1537`

`color(white)(24f(x)) = (12x+7)^2-(sqrt(1537))^2`

`color(white)(24f(x)) = ((12x+7)-sqrt(1537))((12x+7)+sqrt(1537))`

`color(white)(24f(x)) = (12x+7-sqrt(1537))(12x+7+sqrt(1537))`

`color(white)(24f(x)) = 144(x+7/12-sqrt(1537)/12)(x+7/12+sqrt(1537)/12)`

Hence:

`f(x) = 6(x+7/12-sqrt(1537)/12)(x+7/12+sqrt(1537)/12)`