Question

How do you factor the expression `2x^2 +11x-9`?

Answer 1

`2x^2+11x-9 = 1/8(4x+11-sqrt(193))(4x+11+sqrt(193))`

Explanation:

Note that if the `-` sign was a `+`, then this would factor with integer coefficients:

`2x^2+11x+9 = (x+1)(2x+9)`

To factor the given quadratic with a minus sign, we can complete the square, but to cut down on fractions, I would like to multiply through by `8` first, then divide through by `8` at the end...

`8(2x^2+11x-9) = 16x^2+88x-72`

`color(white)(8(2x^2+11x-9)) = (4x)^2+2(11)(4x)+121-193`

`color(white)(8(2x^2+11x-9)) = (4x+11)^2-(sqrt(193))^2`

`color(white)(8(2x^2+11x-9)) = ((4x+11)-sqrt(193))((4x+11)+sqrt(193))`

`color(white)(8(2x^2+11x-9)) = (4x+11-sqrt(193))(4x+11+sqrt(193))`

So:

`2x^2+11x-9 = 1/8(4x+11-sqrt(193))(4x+11+sqrt(193))`