Question

How do you factor `14x ^ { 2} - 7x - 28`?

Answer 1

`14x^2-7x-28 = 7/8(4x-1-sqrt(33))(4x-1+sqrt(33))`

Explanation:

Given:

`f(x) = 14x^2-7x-28`

I will multiply by `8/7` in order to complete the square and use the difference of squares identity while avoiding fractions as much as possible...

`8/7 f(x) = 16x^2-8x-32`

`color(white)(8/7 f(x)) = (4x)^2-2(4x)+1-33`

`color(white)(8/7 f(x)) = (4x-1)^2-(sqrt(33))^2`

`color(white)(8/7 f(x)) = ((4x-1)-sqrt(33))((4x-1)+sqrt(33))`

`color(white)(8/7 f(x)) = (4x-1-sqrt(33))(4x-1+sqrt(33))`

Multiplying both ends by `7/8` we find:

`14x^2-7x-28 = 7/8(4x-1-sqrt(33))(4x-1+sqrt(33))`