Question

How do you factor `7y^2-11y-27`?

Answer 1

`7y^2-11y-27 = 1/28(14y-11-sqrt(877))(14y-11+sqrt(877))`

Explanation:

The difference of squares identity can be written:

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

Use this with `a=(14y-11)` and `b=sqrt(877)`

First premultiply by `7*2^2 = 28`, then divide by it after completing the square:

`28(7y^2-11y-27) = 196y^2-308y-756`

`color(white)(28(7y^2-11y-27)) = (14y)^2-2(14y)(11)+121-877`

`color(white)(28(7y^2-11y-27)) = (14y-11)^2-(sqrt(877))^2`

`color(white)(28(7y^2-11y-27)) = ((14y-11)-sqrt(877))((14y-11)+sqrt(877))`

`color(white)(28(7y^2-11y-27)) = (14y-11-sqrt(877))(14y-11+sqrt(877))`

Hence:

`7y^2-11y-27 = 1/28(14y-11-sqrt(877))(14y-11+sqrt(877))`

Answer 2

(1/28)(14x - 11 - sqrt877)(14x - 11 + sqrt877)

Explanation:

There is another classical way.
f(x) = a(x - x1)(x - x2),
where x1 and x2 are the 2 real roots of the quadratic equation
f(x) = 7x^2 - 11x - 27 = 0.
`D = d^2 = b^2 - 4ac = 121 + 756 = 877` --> `d = +- sqrt877`
There are 2 real roots:
`x = -b/(2a) +- d/(2a) = 11/14 +- sqrt877/14`
`x1 = 11/14 + sqrt877/14`
`x2 = 11/14 - sqrt877/14`
The factored form will be:
`f(x) = (7)(x - 11/14 - sqrt877/14)(x - 11/14 - sqrt877/14) =`
`= 1/28(14x - 11 - sqrt877)(14x - 11 + sqrt877)`