A Cubic System Solved in Two Ways

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a³ + b³ = 2√5 — ①
a²b + ab² = √5 — ②

Multiplying both sides of ② by 3:
3ab(a + b) = 3√5 — ③

Adding ① and ③:
a³ + b³ + 3ab(a + b) = 3√5 + 2√5 = 5√5
(a + b)³ = 5√5
a + b = √5 — ④

From ②:
ab(a + b) = √5

Substituting in ④:
ab√5 = √5
ab = 1
b = 1/a — ⑤
From ④ and ⑤:
a + 1/a = √5

As a ≠ 0 by definiton:
a² + 1 = √5a
a² – √5a + 1 = 0

Using the quadratic fromula:
a = √5/2 ± 1/2

As the equation set is symmetric, b is the conjugate value.
(a, b) = (½√5 ± ½)

[ldcfbht]

Nice method .Thank you .I solved it the second way .Bravo

[mathswan]

a+b=sqrt(5) and ab=1
a and b are the roots of t^2-sqrt(5)t +1=0

[imonkalyanbarua]

I am thoroughly enjoying your channel and watching as many videos as possible. You Sir are amazing and doing a yeoman service to the community of maths enthusiasts! Bravo! 😇🙏

[alnitaka]

At 2:18 you say a+b=s and ab=p. At 4:57 you say that s=sqrt(5) and p = 1. This means s and p are roots of x^2-sqrt(5)x + 1. So my idea is to multiply this by x^2+sqrt(5)x+1 to get x^4-3x^2+1, where we can use the quartic formula. The resolvent cubic is y^3-6y^2+5 = 0 which has roots 0, 1, and 5. So a root of the quartic is = (sqrt(5)+1)/2, the golden ratio. Another root is (1/2)*(-sqrt(0) - sqrt(1)+sqrt(5)) = (sqrt(5)-1)/2. Then continue like you say past 4:57.

[devondevon]

Answer when a=1.61802 b= 0.61802, and when b=1.61802 a=0.61802
This is the Golden ratio
Since (a+b)^3 expands to a^3 + b^3 + 3 (a^2b + b^2 a) then
(a+b)^3 = 2sqrt 5 + 3( sqrt 5) substituting the value of the two equations
= 2 sqrt 5 + 3 sqrt 5
= 5 sqrt 5
=(5^1/2)( 5^1); hence
a + b = (5^1/6)(5^3) cube both sides
= (5^1/6)(5^3)
=5^3/6
a+b = sqrt 5

ab^2 + a^2b = sqrt 5 (given)
ab(a+b) = sqrt 5 factor out 'ab'
ab (sqrt 5)= sqrt 5
ab = sqrt 5/ sqrt 5
ab = 1
So I will use 'ab =1" and 'a+b = sqrt 5" the find the value of "a" and "b"
a = 1/b hence
1/b + b = sqrt 5
1 + b^2 = (sqrt 5)b multiply both sides by 'b"
b^2 - sqrt 5b + 1 =0 using the quadratic formulae gives
b = 1.61802 and b= 0.61802 ANSWER
a= 1/1.61802 = 0.61802 ANSWER
a= 1/0.61802 = 1.61802 ANSWER

[mahmoudalbahar]

Many thanks to you .

this is very much (cool+clever).

[jamescollis]

Nice system. Would there be complex solutions also?

[zana]

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