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Algebra Level 4

Find the sum of all real solutions of the equation:

x 4 + 8 x 2 9 + 1 = 3 x 3 + 3 x . x^{4} + \frac{8x^{2}}{9} + 1 = 3x^{3} + 3x.


The answer is 3.34.

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Aditya Tiwari by Aditya Tiwari , 22, India

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1 solution

Atul Solanki
Dec 31, 2014

arrange all terms as 9 x 4 − 27 x 3 + 8 x 2 − 27 x + 9 = 0 9x^4-27x^3+8x^2-27x+9=0 now divide by x 2 x^2 => 9 x 2 − 27 x + 8 − 27 / x + 9 / x 2 = 0 9x^2-27x+8-27/x+9/x^2=0 => 9 ( x 2 + 1 / x 2 ) − 27 ( x + 1 / x ) + 8 = 0 9(x^2+1/x^2)-27(x+1/x)+8=0 let x + 1 / x = t x+1/x=t => x 2 + 1 / x 2 = t 2 − 2 x^2+1/x^2=t^2-2 => 9 t 2 − 27 t − 10 = 0 9t^2-27t-10=0 => ( 3 t + 1 ) ( 3 t − 10 ) = 0 (3t+1)(3t-10)=0 now for t = − 1 / 3 t=-1/3 doesn't give real value of x => t cannot be equal to − 1 / 3 -1/3 => t = 10 / 3 t=10/3 => x = 3 , 1 / 3 x=3,1/3 => 3 + 1 / 3 3+1/3 =3.33]

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