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)
