Answer:
1. x = -2 or x = sqrt(6) - 2 or x = -2 - sqrt(6)
2. x = -2.10947 or x = -0.484343 or x = 1.67884 or x = 2.91497
Explanation:
Solve for x:
x^3 + 6 x^2 + 6 x - 4 = 0
The left hand side factors into a product with two terms:
(x + 2) (x^2 + 4 x - 2) = 0
Split into two equations:
x + 2 = 0 or x^2 + 4 x - 2 = 0
Subtract 2 from both sides:
x = -2 or x^2 + 4 x - 2 = 0
Add 2 to both sides:
x = -2 or x^2 + 4 x = 2
Add 4 to both sides:
x = -2 or x^2 + 4 x + 4 = 6
Write the left hand side as a square:
x = -2 or (x + 2)^2 = 6
Take the square root of both sides:
x = -2 or x + 2 = sqrt(6) or x + 2 = -sqrt(6)
Subtract 2 from both sides:
x = -2 or x = sqrt(6) - 2 or x + 2 = -sqrt(6)
Subtract 2 from both sides:
Answer: x = -2 or x = sqrt(6) - 2 or x = -2 - sqrt(6)
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Solve for x:
x^4 - 2 x^3 - 6 x^2 + 8 x + 5 = 0
Eliminate the cubic term by substituting y = x - 1/2:
5 + 8 (y + 1/2) - 6 (y + 1/2)^2 - 2 (y + 1/2)^3 + (y + 1/2)^4 = 0
Expand out terms of the left hand side:
y^4 - (15 y^2)/2 + y + 117/16 = 0
Subtract -3/2 sqrt(13) y^2 - (15 y^2)/2 + y from both sides:
y^4 + (3 sqrt(13) y^2)/2 + 117/16 = (3 sqrt(13) y^2)/2 + (15 y^2)/2 - y
y^4 + (3 sqrt(13) y^2)/2 + 117/16 = (y^2 + (3 sqrt(13))/4)^2:
(y^2 + (3 sqrt(13))/4)^2 = (3 sqrt(13) y^2)/2 + (15 y^2)/2 - y
Add 2 (y^2 + (3 sqrt(13))/4) λ + λ^2 to both sides:
(y^2 + (3 sqrt(13))/4)^2 + 2 λ (y^2 + (3 sqrt(13))/4) + λ^2 = -y + (3 sqrt(13) y^2)/2 + (15 y^2)/2 + 2 λ (y^2 + (3 sqrt(13))/4) + λ^2
(y^2 + (3 sqrt(13))/4)^2 + 2 λ (y^2 + (3 sqrt(13))/4) + λ^2 = (y^2 + (3 sqrt(13))/4 + λ)^2:
(y^2 + (3 sqrt(13))/4 + λ)^2 = -y + (3 sqrt(13) y^2)/2 + (15 y^2)/2 + 2 λ (y^2 + (3 sqrt(13))/4) + λ^2
-y + (3 sqrt(13) y^2)/2 + (15 y^2)/2 + 2 λ (y^2 + (3 sqrt(13))/4) + λ^2 = (2 λ + 15/2 + (3 sqrt(13))/2) y^2 - y + (3 sqrt(13) λ)/2 + λ^2:
(y^2 + (3 sqrt(13))/4 + λ)^2 = y^2 (2 λ + 15/2 + (3 sqrt(13))/2) - y + (3 sqrt(13) λ)/2 + λ^2
Complete the square on the right hand side:
(y^2 + (3 sqrt(13))/4 + λ)^2 = (y sqrt(2 λ + 15/2 + (3 sqrt(13))/2) - 1/(2 sqrt(2 λ + 15/2 + (3 sqrt(13))/2)))^2 + (4 (2 λ + 15/2 + (3 sqrt(13))/2) (λ^2 + (3 sqrt(13) λ)/2) - 1)/(4 (2 λ + 15/2 + (3 sqrt(13))/2))
To express the right hand side as a square, find a value of λ such that the last term is 0.
This means 4 (2 λ + 15/2 + (3 sqrt(13))/2) (λ^2 + (3 sqrt(13) λ)/2) - 1 = 8 λ^3 + 18 sqrt(13) λ^2 + 30 λ^2 + 45 sqrt(13) λ + 117 λ - 1 = 0.
Thus the root λ = 1/4 (-3 sqrt(13) - 5) + (2 2^(2/3) (i sqrt(3) + 1))/(i sqrt(183) - 29)^(1/3) + ((-i sqrt(3) + 1) (i sqrt(183) - 29)^(1/3))/(2 2^(2/3)) allows the right hand side to be expressed as a square.
(This value will be substituted later):
(y^2 + (3 sqrt(13))/4 + λ)^2 = (y sqrt(2 λ + 15/2 + (3 sqrt(13))/2) - 1/(2 sqrt(2 λ + 15/2 + (3 sqrt(13))/2)))^2
Take the square root of both sides:
y^2 + (3 sqrt(13))/4 + λ = y sqrt(2 λ + 15/2 + (3 sqrt(13))/2) - 1/(2 sqrt(2 λ + 15/2 + (3 sqrt(13))/2)) or y^2 + (3 sqrt(13))/4 + λ = -y sqrt(2 λ + 15/2 + (3 sqrt(13))/2) + 1/(2 sqrt(2 λ + 15/2 + (3 sqrt(13))/2))
Solve using the quadratic formula:
y = 1/4 (sqrt(2) sqrt(4 λ + 15 + 3 sqrt(13)) + sqrt(2) sqrt((108 - 24 sqrt(13) λ - 16 λ^2 - 4 sqrt(2) sqrt(4 λ + 15 + 3 sqrt(13)))/(4 λ + 15 + 3 sqrt(13)))) or y = 1/4 (sqrt(2) sqrt(4 λ + 15 + 3 sqrt(13)) - sqrt(2) sqrt((108 - 24 sqrt(13) λ - 16 λ^2 - 4 sqrt(2) sqrt(4 λ + 15 + 3 sqrt(13)))/(4 λ + 15 + 3 sqrt(13)))) or y = 1/4 (sqrt(2) sqrt((108 - 24 sqrt(13) λ - 16 λ^2 + 4 sqrt(2) sqrt(4 λ + 15 + 3 sqrt(13)))/(4 λ + 15 + 3 sqrt(13))) - sqrt(2) sqrt(4 λ + 15 + 3 sqrt(13))) or y = 1/4 (-sqrt(2) sqrt(4 λ + 15 + 3 sqrt(13)) - sqrt(2) sqrt((108 - 24 sqrt(13) λ - 16 λ^2 + 4 sqrt(2) sqrt(4 λ + 15 + 3 sqrt(13)))/(4 λ + 15 + 3 sqrt(13)))) where λ = 1/4 (-3 sqrt(13) - 5) + (2 2^(2/3) (i sqrt(3) + 1))/(i sqrt(183) - 29)^(1/3) + ((-i sqrt(3) + 1) (i sqrt(183) - 29)^(1/3))/(2 2^(2/3))
Substitute λ = 1/4 (-3 sqrt(13) - 5) + (2 2^(2/3) (i sqrt(3) + 1))/(i sqrt(183) - 29)^(1/3) + ((-i sqrt(3) + 1) (i sqrt(183) - 29)^(1/3))/(2 2^(2/3)) and approximate:
y = -2.60947 or y = -0.984343 or y = 1.17884 or y = 2.41497
Substitute back for y = x - 1/2:
x - 1/2 = -2.60947 or y = -0.984343 or y = 1.17884 or y = 2.41497
Add 1/2 to both sides:
x = -2.10947 or y = -0.984343 or y = 1.17884 or y = 2.41497
Substitute back for y = x - 1/2:
x = -2.10947 or x - 1/2 = -0.984343 or y = 1.17884 or y = 2.41497
Add 1/2 to both sides:
x = -2.10947 or x = -0.484343 or y = 1.17884 or y = 2.41497
Substitute back for y = x - 1/2:
x = -2.10947 or x = -0.484343 or x - 1/2 = 1.17884 or y = 2.41497
Add 1/2 to both sides:
x = -2.10947 or x = -0.484343 or x = 1.67884 or y = 2.41497
Substitute back for y = x - 1/2:
x = -2.10947 or x = -0.484343 or x = 1.67884 or x - 1/2 = 2.41497
Add 1/2 to both sides:
Answer: x = -2.10947 or x = -0.484343 or x = 1.67884 or x = 2.91497