a + b = 2
a^4 + b^4 = 56
b = 2 - a
Sub b into the 2nd equation.
Solve for a:
(2 - a)^4 + a^4 = 56
Expand out terms of the left hand side:
2 a^4 - 8 a^3 + 24 a^2 - 32 a + 16 = 56
Subtract 56 from both sides:
2 a^4 - 8 a^3 + 24 a^2 - 32 a - 40 = 0
The left hand side factors into a product with three terms:
2 (a^2 - 2 a - 2) (a^2 - 2 a + 10) = 0
Divide both sides by 2:
(a^2 - 2 a - 2) (a^2 - 2 a + 10) = 0
Split into two equations:
a^2 - 2 a - 2 = 0 or a^2 - 2 a + 10 = 0
Add 2 to both sides:
a^2 - 2 a = 2 or a^2 - 2 a + 10 = 0
Add 1 to both sides:
a^2 - 2 a + 1 = 3 or a^2 - 2 a + 10 = 0
Write the left hand side as a square:
(a - 1)^2 = 3 or a^2 - 2 a + 10 = 0
Take the square root of both sides:
a - 1 = sqrt(3) or a - 1 = -sqrt(3) or a^2 - 2 a + 10 = 0
Add 1 to both sides:
a = 1 + sqrt(3) or a - 1 = -sqrt(3) or a^2 - 2 a + 10 = 0
Add 1 to both sides:
a = 1 + sqrt(3) or a = 1 - sqrt(3) or a^2 - 2 a + 10 = 0
Subtract 10 from both sides:
a = 1 + sqrt(3) or a = 1 - sqrt(3) or a^2 - 2 a = -10
Add 1 to both sides:
a = 1 + sqrt(3) or a = 1 - sqrt(3) or a^2 - 2 a + 1 = -9
Write the left hand side as a square:
a = 1 + sqrt(3) or a = 1 - sqrt(3) or (a - 1)^2 = -9
Take the square root of both sides:
a = 1 + sqrt(3) or a = 1 - sqrt(3) or a - 1 = 3 i or a - 1 = -3 i
Add 1 to both sides:
a = 1 + sqrt(3) or a = 1 - sqrt(3) or a = 1 + 3 i or a - 1 = -3 i
Add 1 to both sides:
a = 1 + sqrt(3) or a = 1 - sqrt(3) or a = 1 + 3 i or a = 1 - 3 i
