Use completion of squares to determine the root of the equation 3x^2 - 6x + 8 = 0

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Use the quadratic formula to find the solutions. In this case, the values are `a=3` , `b=-6` , and `c=8` .

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

` `where


  `ax^2+bx+c=0`


Substitute in the values of `a=3` , `b=-6` , and `c=8` .

`x=(-(-6)+- sqrt((-6)^2-4(3)(8)))/(2(3))`


Multiply `-1` by each term inside the parentheses.

`x=(6+- sqrt((-6)^2-4(3)(8)))/(2(3))`


Simplify the section inside the radical.

`x=(6+-2i sqrt(15))/(2(3))`

Simplify the denominator of the quadratic formula.

`x=(6+-2i sqrt(15))/6`

Simplify the expression to solve for the `+` portion of the `+-.`

`x=(6+2i sqrt(15))/6`

Simplify the expression to solve for the` -` portion of the `+-.`

`x=(6-2i sqrt(15))/6`


The final answer is the combination of both solutions

`x=(6+2i sqrt(15))/6, (6-2i sqrt(15))/6`

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The quadratic equation 3x^2 - 6x + 8 = 0 has to be solved by the method completing the square.

The method relies on the identity `a^2 + 2a*b + b^2 = (a+b)^2`

For the given equation:

`3x^2 - 6x + 8 = 0`

Divide both the sides of the equation by 3.

`x^2 - 2x + 8/3 = 0`

=> `x^2 - 2*1*x + 1^2 = 1 - 8/3`

=> `(x - 1)^2 = -5/3`

`x - 1 = +-sqrt(-5/3)`

=> `x = 1 +- sqrt(-5/3)`

=> `x = 1 +- i*sqrt(5/3)`

The solution of the equation `x^2 - 2x + 8/3 = 0` is `x = 1 +- i*sqrt(5/3)`

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