solve by completing the square: x^2 + 1/2x = 1/3?

x^2 + (1/2)x + [(1/2)/2]^2 = 1/3 + [(1/2)/2]^2
x^2 + (1/2)x + (1/4)^2 = 1/3 + (1/4)^2
(x + 1/4)^2 = 1/3 + 1/16
(x + 1/4)^2 = 19/48
x + 1/4 = sqrt(19/48) or -sqrt(19/48)
x = -0.879 or 0.379 (correct to 3 significant figures)

x ² + (1/2) x + 1/16 = 1 / 3 + 1 / 16
(x + 1/4)² = 16 / 48 + 3 / 48
(x + 1/4)² = 19 / 48
(x + 14) = ± √ (19 / 48)
x = - 14 ± √ (19 / 48)
x = - 14 ± √ (19 ) / (4√3)
x = - 14 ± √ (57 ) / 12

x^2 + 1/2(x) = 1/3
x^2 + x/4 + x/4 = 1/3
x^2 + x/4 + x/4 + 1/16 = 1/3 + 1/16
(x^2 + x/4) + (x/4 + 1/16) = 16/48 + 3/48
x(x + 1/4) + 1/4(x + 1/4) = 19/48
(x + 1/4)^2 = 19/48
x + 1/4 = ±√(19/48)
x = ±√(19/48) - 1/4

x^2 + 1/2x = 1/3

x^2 + 1/2x + 1/16 = 1/3 + 1/16

(x + 1/4)^2 = 19/48

x + 1/4 = + or - sqrt(19/48)

so you get two answers:

x = -1/4 + sqrt(19/48) and x = -1/4 - sqrt(19/48)

there you go.

to complete the square you add (1/2 of the coefficient of x)^2 on both sides:
(a + b)^2 = a^2 + 2ba + b^2
so if you halve the number in front of a, you get b and then if you square b, you get b^2, and if you add this to a^2 + 2ba, you have completed the square.

x^2 + (1/2)x = 1/3
x^2 + (1/2)x + (1/4)^2 = 1/3 + 1/16 [add (1/4)^2 on both sides and halve of (1/2) is (1/4) and then you square it]
(x + 1/4)^2 = 1/3 + 1/16
(x + 1/4)^2 = 19/48
x + 1/4 = sqrt (19/48) or -sqrt (19/48) (square root both sides)
x = -1/4 + sqrt (19/48) or - 1/4 -sqrt (19/48)