+0  
 
0
85
2
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Solve

 

a + b = 2

a^4 + b^4 = 56

 

in real numbers.

 Jun 7, 2020
 #1
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+3

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

 Jun 7, 2020
 #2
avatar+110671 
0

You have done a lot of work there guest!

 

Good going. I'd give you more points if I could.        laugh cool laugh

 

(I have not checked it properly but it looks impressive)   wink

 Jun 8, 2020

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