Question

How do you solve `18u^2-3u=1`?

Answer 1

`u=1/3` or `u= -1/6`

Explanation:

Method 1
Solutions for `18u^2-3u = 1` or equivalently `18u^2-3u -1 =0`
can be determined using the quadratic formula (see bottom, if you don't know this formula).

`u= (-(-3)+-sqrt((-3)^2-4(18)(-1)))/(2(18))`

`color(white)("XXXX")` `= (3+-sqrt(81))/(36)`

`u = 12/36 = 1/3`
or
`u=-6/36 = -1/6`

Method 2
If we recognize that (after moving the 1 to the left side) we can factor:
`color(white)("XXXX")` `18y^2-3u-1=0`
to get
`color(white)("XXXX")` `(3u-1)(6u+1)=0`

Then either
`color(white)("XXXX")` `(3u-1) = 0` which implies `u=1/3`
or
`color(white)("XXXX")` `(6u+1)=0` which implies `u=-1/6`

Answer 2

Factor `y = 18x^2 - 3x - 1`

Explanation:

`y = 18x^2 - 3x - 1 = 18(x - p)(x - q).`
I use the new AC Method (Google, Yahoo Search)
Converted `y' = x^2 - 3x - 18` = (x - p')(x - q').
Factor pairs of (-18)--> (-2, 9)(-3, 6). This sum is 3 = -b
p' = 3 and q' = - 6
`p = 3/18 = 1/6` and `q = -6/18 = - 1/3`

Factored form: y = 18(x + 1/6)(x - 1/3) =`(6x + 1)(3x - 1)`