3x2 - 2x - 6

Question

find the roots of the polynomial

Answer 1

Let 3x²-2x-6=0, then the solution to this equation will be the roots.

3x²-2x=6,

x²-⅔x=2,

x²-⅔x+1/9=2+1/9,

(x-⅓)²=19/9,

(x-⅓)²-19/9=0. Now we factorise the difference of two squares: a²-b²=(a-b)(a+b):

(x-⅓)²-(√19/3)²=(x-⅓-⅓√19)(x-⅓+⅓√19)=0,

so x-⅓-⅓√19=0⇒x=⅓(1+√19); or:

x-⅓+⅓√19=0⇒x=⅓(1-√19). 

The same result is obtained using the Quadratic Formula.

The original expression can be written:

3(x-⅓-⅓√19)(x-⅓+⅓√19) or (3x-1-√19)(x-⅓+⅓√19).