Vietas?

Algebra Level 3

If x 1 x_1 and x 2 x_2 are the roots of the equation x 2 − 6 x + 4 = 0 x^2-6x+4=0 , then ( x 1 − 2 ) 16 + ( x 2 − 6 ) 8 x 1 8 \dfrac{(x_1-2)^{16} + (x_2-6)^{8}}{x_1^8} is equal to


The answer is 257.

Vilakshan Gupta by Vilakshan Gupta , 19, India

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1 solution

Vilakshan Gupta
Aug 29, 2017

The given expression can be written as ( x 1 − 2 ) 16 x 1 8 + ( x 2 − 6 ) 8 x 1 8 \dfrac{(x_1-2)^{16}}{x_1^8} + \dfrac{(x_2-6)^8}{x_1^8} By Vieta's Formula, x 1 + x 2 = 6 x_1+x_2=6 ⟹ \implies x 2 = 6 − x 1 x_2=6-x_1 . Now putting the value of x 2 x_2 in the fraction and rewriting the equation, ( ( x 1 − 2 ) 2 x 1 ) 8 + ( − x 1 x 1 ) 8 = ( ( x 1 2 − 4 x 1 + 4 ) x 1 ) 8 + 1 = ( x 1 − 4 + 4 x 1 ) 8 + 1 \large \left(\frac{(x_1-2)^2}{x_1}\right)^8 + \left(\frac{-x_1}{x_1}\right)^8 \\ = \left(\frac{(x_1^2-4x_1+4)}{x_1}\right)^8 + 1 \\ = \left(x_1-4+\frac{4}{x_1}\right)^8 + 1 Now again by Vieta's Formula x 1 x 2 = 4 x_1x_2=4 , ∴ \therefore The equation changes to ( x 1 + x 2 − 4 ) 8 + 1 = ( 6 − 4 ) 8 + 1 (x_1+x_2-4)^8+1 \\ = (6-4)^8+1 therefore final answer is 2 8 + 1 = 257 2^8+1=\color{#3D99F6}\boxed{257}

3 Helpful 0 Interesting 0 Brilliant 0 Confused

Yup!!! Same way!!

Aaghaz Mahajan - 3 years ago

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