Solve for x:
x^4 + 10 x + 25 = 0
Subtract 10 x - 10 x^2 from both sides:
x^4 + 10 x^2 + 25 = 10 x^2 - 10 x
x^4 + 10 x^2 + 25 = (x^2 + 5)^2:
(x^2 + 5)^2 = 10 x^2 - 10 x
Add 2 (x^2 + 5) λ + λ^2 to both sides:
(x^2 + 5)^2 + 2 λ (x^2 + 5) + λ^2 = -10 x + 10 x^2 + 2 λ (x^2 + 5) + λ^2
(x^2 + 5)^2 + 2 λ (x^2 + 5) + λ^2 = (x^2 + 5 + λ)^2:
(x^2 + 5 + λ)^2 = -10 x + 10 x^2 + 2 λ (x^2 + 5) + λ^2
-10 x + 10 x^2 + 2 λ (x^2 + 5) + λ^2 = (2 λ + 10) x^2 - 10 x + 10 λ + λ^2:
(x^2 + 5 + λ)^2 = x^2 (2 λ + 10) - 10 x + 10 λ + λ^2
Complete the square on the right hand side:
(x^2 + 5 + λ)^2 = (x sqrt(2 λ + 10) - 5/sqrt(2 λ + 10))^2 + (4 (2 λ + 10) (λ^2 + 10 λ) - 100)/(4 (2 λ + 10))
Solve using the quadratic formula:
x = 1/2 (sqrt(2) sqrt(λ + 5) + sqrt(2) sqrt(-(25 + 10 λ + λ^2 + 5 sqrt(2) sqrt(λ + 5))/(λ + 5))) or x = 1/2 (sqrt(2) sqrt(λ + 5) - sqrt(2) sqrt(-(25 + 10 λ + λ^2 + 5 sqrt(2) sqrt(λ + 5))/(λ + 5))) or x = 1/2 (sqrt(2) sqrt((-25 - 10 λ - λ^2 + 5 sqrt(2) sqrt(λ + 5))/(λ + 5)) - sqrt(2) sqrt(λ + 5)) or x = 1/2 (-sqrt(2) sqrt(λ + 5) - sqrt(2) sqrt((-25 - 10 λ - λ^2 + 5 sqrt(2) sqrt(λ + 5))/(λ + 5))) where λ = -5 + (5 2^(2/3))/(3/5 (i sqrt(1119) + 9))^(1/3) + (5/6)^(2/3) (i sqrt(1119) + 9)^(1/3)
Substitute λ = -5 + (5 2^(2/3))/(3/5 (i sqrt(1119) + 9))^(1/3) + (5/6)^(2/3) (i sqrt(1119) + 9)^(1/3) and approximate:
x = -1.61762 - 1.035 i or x = -1.61762 + 1.035 i or x = 1.61762 - 2.04014 i or x = 1.61762 + 2.04014 i
