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Doubt on solving equations

If \(a,b,c\) are real numbers and \(a+b+c\)=M , \(a^{3}+b^{3}+c^{3}=M^{3}\) \[\sum_{cyc}^{a,b,c}\frac{a^{3}}{1-a^{2}}=\frac{1}{6}\] . Then find M. Can you please help me to solve this.

Note by Shivam Jadhav
1 year, 7 months ago

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By given two conditions we get 3(a+b)(b+c)(c+a)=0
hence we can write a=-b and our summation reduces to c^3/1-c^2
Hence we find one of roots of this cubic as 0.5.
Hence M =0.5 Aakash Khandelwal · 1 year, 7 months ago

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@Aakash Khandelwal Can you please elaborate? Shivam Jadhav · 1 year, 7 months ago

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\((a + b + c)^3 = a^3 + b^3 + c^3 + 3(a + b)(b + c)(c + a)\)

a + b + c = M

\(a^3 + b^3 + c^3 = M^3\)

which is a = b = c = 0 Dev Sharma · 1 year, 7 months ago

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@Dev Sharma also a + b = 0

b + c = 0

c + a = 0 Dev Sharma · 1 year, 7 months ago

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