2x^2+4x+1 the quadratic and completing the square

Quadratic Formula:

Solve for x over the real numbers:2 x^2 + 4 x + 1 = 0

x = (-4 ± sqrt (4^2 - 4*2)) / (2*2) = (-4 ± sqrt (16 - 8)) / 4 = (-4 ± sqrt (8)) / 4:x = (-4 + sqrt (8)) / 4 or x = (-4 - sqrt (8)) / 4

sqrt (8) = sqrt (2^3) = 2sqrt (2) : x = (-4 + 2 sqrt (2)) / 4 or x = (-4 - 2 sqrt (2)) / 4

Factor 2 from - 4 + 2 sqrt (2) giving 2 (sqrt (2) - 2) : x = (2 (sqrt (2) - 2)) / 4 or x = (-2 sqrt (2) - 4) / (4)

(2 (sqrt (2) - 2)) / 4 = (2 (sqrt (2) - 2)) / (2*2) = (sqrt (2) - 2) / 2:x = (sqrt (2) - 2) / 2 or x = (-2 sqrt (2) - 4) / (4)

Factor 2 from - 4 - 2 sqrt (2) giving 2 (-sqrt (2) - 2) : x = 1/2 (sqrt (2) - 2) or x = (2 (-sqrt (2) - 2)) / 4

(2 (-sqrt (2) - 2)) / 4 = (2 (-sqrt (2) - 2)) / (2*2) = (-sqrt (2) - 2) / 2:Answer: x = 1/2 (sqrt (2) - 2) or x = (-sqrt (2) - 2) / 2

Complete the Square:

Solve for x over the real numbers:2 x^2 + 4 x + 1 = 0

Divide both sides by 2:x^2 + 2 x + 1/2 = 0

Subtract 1/2 from both sides:x^2 + 2 x = - 1/2

Add 1 to both sides:x^2 + 2 x + 1 = 1/2

Write the left-hand side as a square: (x + 1) ^2 = 1/2

Take the square root of both sides:x + 1 = 1/sqrt (2) or x + 1 = - 1/sqrt (2)

Subtract 1 from both sides:x = 1/sqrt (2) - 1 or x + 1 = - 1/sqrt (2)

Subtract 1 from both sides:Answer: x = 1/sqrt (2) - 1 or x = - 1 - 1/sqrt (2)