We can use the quadratic formula to solve this.
#3x^2 - x- 1 = 0#
#ax^2 + bx + c = 0#
#a = 3#
#b = -1#
#c = -1#
# x = (-b +- sqrt(b^2-4ac)) / (2a) #
# x = (-(-1) +- sqrt(-1^2-4 xx 3 xx -1)) / (2 xx 6) #
# x = (1 +- sqrt( 1 - 12 xx -1)) / (2 xx 6) #
# x = (1 +- sqrt( 1 - -12)) / 12 #
# x = (1 +- sqrt13) / 12 #
# x = (1 +- 3.605551275) / 12 #
# x_1 = (1 + 3.605551275) / 12 #
# x_1 = (4.605551275) / 12 #
#color(blue)( x_1 = 0.3837959396 or (1 + sqrt13) / 12 #
# x_2 = (1 - 3.605551275) / 12 #
# x_2 = -2.605551275 / 12 #
#color(blue)( x_2 = -0.217129273 or (1 - sqrt13) / 12 #