Ta có:
` a + b + c = 0 `
` <=> (a + b + c)^2 = 0 `
` <=> a^2 + b^2 + c^2 + 2ab + 2bc + 2ca = 0 `
Vì ` a^2 + b^2 + c^2 = x^2 <=> 2ab + 2bc + 2ca = -(x^{2}) `
` <=> 2(ab + bc + ca) = -(x^2) `
` <=> ab + bc + ca = -\frac{x^2}{2} `
` <=> (ab + bc + ca)^2 = \frac{x^4}{4} `
` <=> a^{2}b^{2} + b^{2}c^{2} + c^{2}a^{2} + 2abc(a + b + c) = \frac{x^4}{4} `
` <=> a^{2}b^{2} + b^{2}c^{2} + c^{2}a^{2} = \frac{x^4}{4} `
Mặt khác:
` a^2 + b^2 + c^2 = x^2 `
` <=> (a^2 + b^2 + c^2)^2 = x^4 `
` <=> a^4 + b^4 + c^4 + 2a^{2}b^{2} + 2b^{2}c^{2} + 2c^{2}a^{2} = x^4 `
` <=> a^4 + b^4 + c^4 + 2(a^{2}b^{2} + b^{2}c^{2} + c^{2}a^{2}) = x^4 `
` <=> a^4 + b^4 + c^4 + \frac{2x^{4}}{4} = x^4 `
` <=> a^4 + b^4 + c^4 = \frac{4x^4}{4} - \frac{2x^4}{4} `
` <=> a^4 + b^4 + c^4 = \frac{x^4}{2} `
