Question

How do you solve `9x ^ { 2} - 36x =6`?

Answer 1

`x =(36 +- 6sqrt42) / (18)`

Explanation:

`9x^2 - 36x = 6`

We can move `6` to the other side, then use the quadratic formula.

`9x^2 - 36x -6 = 0`

`ax^2 + bx +c = 0`

`a = 9`
`b = -36`
`c = -6`

`x = (-b +- sqrt(b^2-4ac)) / (2a)`

`x = (- -36 +- sqrt((-36)^2-4 xx 9 xx -6)) / (2 xx 9)`

Now we can simplify and solve.

`x = (cancel(- -)36 +- sqrt(-36^2-4 xx 9 xx -6)) / (18)`

`x =(36 +- sqrt((-36)^2- 36 xx -6)) / (18)`

`x =(36 +- sqrt((-36)^2 + 216)) / (18)`

`x =(36 +- sqrt(1296 + 216)) / (18)`

`x =(36 +- sqrt(1512)) / (18)`

`color(blue)( x =(36 +- 6sqrt42) / (18)`

`therefore` `x =(36 +- sqrt(1512)) / (18)`

Answer 2

`x = (6 +- sqrt42)/3`

Explanation:

`y = 9x^2 - 36x - 6 = 0`
`y = 3(3x^2 - 12x - 2) = 0`
Use the improved quadratic formula (Google, Socratic Search):
`D = d^2 = b^2 - 4ac = 144 + 24 = 168` --> `d = +- 2sqrt42`
There are 2 real roots:
`x = -b/(2a) +- d/(2a) = 12/6 +- (2sqrt42)/6 = 2 +- sqrt42/3 =`
`x = (6 +- sqrt42)/3`