l 3x^2 - 9x +4 l = 14

We have two cases:

Case(1);

3x^2 - 9x + 4 = 14

==> 3x^2 - 9x +4 -14 = 0

==> 3x^2 - 9x -10 = 0

Now we will use the quadratic equation to solve for x.

==> x1 = ( 9 + sqrt(81+4*3*10) / 6

= ( 9 + sqrt(201) / 6

**==> x1= ( 3/2) + sqrt(210) / 6**

**==> x2= (3/2) - sqrt(210) / 6**

** **

Case(2).

==> -(3x^2 -9x +4 ) = 14

==> -3x^2 + 9x -4 = 14

==> -3x^2 + 9x -18 = 0

==> x1= ( -9 + sqrt(81-4*3*18) / -6

= ( -9 + sqrt(135)*i ) / -6

= ( 3/2) + 3sqrt(15) i /6

= (3/2) + (sqrt15 /2 )*i

**==> x1= (3/2) + (sqrt15 / 2) *i**

**==> x2 = (3/2) - 9sqrt15 /2)*i**

The given equation | 3x^2 – 9x + 4| = 14 is equivalent to the following equations:

- 3x^2 – 9x + 4 = 14

=> 3x^2 – 9x – 10 = 0

x1 = [-b + sqrt (b^2 – 4ac)]/2a

=> [9 + sqrt (81 + 120)]/ 6

=> (9 + sqrt 201)/6

x2 = (9 - sqrt 201)/6

- 3x^2 – 9x + 4 = -14

=> 3x^2 – 9x + 18 = 0

x1 = [-b + sqrt (b^2 – 4ac)]/2a

=> [9 + sqrt (-135)]/6

=> (9 + i*sqrt 135) / 6

x2 = (9 - i*sqrt 135) / 6

**Therefore the equation has the roots : (9 + sqrt 201)/6 , x2 = (9 - sqrt 201)/6 , (9 + i*sqrt 135)/6 and (9 - i*sqrt 135) / 6**