Find the Value of the Expression

Algebra Level 2

If 3 x 2 + x = 1 3x^2+x=1 , find 10 x 3 + 27 x 6 10x^3+27x^6 .


The answer is 1.

Aly Ahmed by Aly Ahmed , 34, Egypt

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2 solutions

3 x 2 + x = 1 ⟹ 3 x 2 = 1 − x ⟹ 27 x 6 + 10 x 3 = ( 1 − x ) 3 + 10 x 3 = 1 − 3 x + 3 x ( 3 x 2 + x ) = 1 − 3 x + 3 x × 1 = 1 3x^2+x=1\implies 3x^2=1-x\implies 27x^6+10x^3=(1-x)^3+10x^3=1-3x+3x(3x^2+x)=1-3x+3x\times 1=\boxed 1 .

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Chew-Seong Cheong
Mar 26, 2020

3 x 2 + x = 1 Cube both sides 27 x 6 + 27 x 5 + 9 x 4 + x 3 = 1 27 x 6 + 9 x 3 ( 3 x 2 + x ) + x 3 = 1 Note that 3 x 2 + x = 1 27 x 6 + 9 x 3 ( 1 ) + x 3 = 1 10 x 3 + 27 x 6 = 1 \begin{aligned} 3x^2 + x & = 1 & \small \blue{\text{Cube both sides}} \\ 27x^6 + 27x^5 + 9x^4 + x^3 & = 1 \\ 27x^6 + 9x^3(\blue{3x^2+x}) + x^3 & = 1 & \small \blue{\text{Note that }3x^2+x = 1} \\ 27x^6 + 9x^3(\blue 1) + x^3 & = 1 \\ 10x^3 + 27x^6 & = \boxed 1 \end{aligned}

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