What are the possible values of x for the equation 3x² – x – 1 = 0?

To solve the equation ( 3x² – x – 1 = 0 ), we can use the quadratic formula:

[

x = frac{-b pm sqrt{b^2 – 4ac}}{2a}

]

In this case, ( a = 3 ), ( b = -1 ), and ( c = -1 ).

First, we calculate the discriminant ( b^2 – 4ac ):

[

(-1)^2 – 4(3)(-1) = 1 + 12 = 13

]

Now, we can apply the quadratic formula:

[

x = frac{-(-1) pm sqrt{13}}{2(3)} = frac{1 pm sqrt{13}}{6}

]

This gives us the solutions:

[

x = frac{1 – sqrt{13}}{6} quad text{or} quad x = frac{1 + sqrt{13}}{6}

]

Looking at the options provided, the correct answers are:

– ( x = 1 – frac{sqrt{13}}{6} )

– ( x = 1 + frac{sqrt{13}}{6} )

So, the correct answer is:

x = 1 – √13/6 or x = 1 + √13/6.