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Algebra Level 3

If x = 3 + 1 2 x = \dfrac{\sqrt 3 + 1}2 , find the value of 4 x 3 + 2 x 2 − 8 x + 7 4x^3 + 2x^2 - 8x + 7 .

30 10 15 3 None of these 5

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Saraswat Bhattacharya by Saraswat Bhattacharya , 19, India

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1 solution

Chew-Seong Cheong
Mar 11, 2016

If x = 3 + 1 2 x = \dfrac{\sqrt{3}+1}{2} then x x is a root of 2 x 2 − 2 x − 1 = 0 2x^2 - 2x - 1 = 0 . Therefore, 2 x 2 = 2 x + 1 2x^2 = 2x + 1 . Then, we have:

4 x 3 + 2 x 2 − 8 x + 7 = 2 x ( 2 x + 1 ) + ( 2 x + 1 ) − 8 x + 7 = 4 x 2 + 2 x + 2 x − 8 x + 8 = 2 ( 2 x + 1 ) − 4 x + 8 = 4 x + 2 − 4 x + 8 = 10 \begin{aligned} 4x^3 + 2x^2 - 8x + 7 & = 2x(2x+1) + (2x+1) - 8x + 7 \\ & = 4x^2 + 2x + 2x - 8x +8 \\ & = 2(2x+1) -4x+8 \\ & = 4x + 2 - 4x + 8 \\ & = \boxed{10} \end{aligned}

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