Substitution is blind - 4

Algebra Level 2

If x = 3 − 1 x = \sqrt{3} - 1 then find the value of x 4 + 4 x 3 + 6 x 2 + 4 x − 7 x^4 + 4x^3 + 6x^2 + 4x - 7 .

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The answer is 1.

Ram Mohith by Ram Mohith , 18, India

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2 solutions

The given expression is equal to ( x + 1 ) 4 − 8 = ( √ 3 ) 4 − 8 = 9 − 8 = 1 (x+1)^4-8=(√3)^4-8=9-8=1

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Ram Mohith
Oct 8, 2019

x + 1 = 3 ⟹ x 2 + 2 x + 1 = 3 ⟹ x 2 + 2 x = 2 x + 1 = \sqrt{3} \implies x^2 + 2x + 1 = 3 \implies {\color{#E81990}x^2 + 2x = 2}

( x 2 + 2 x ) 2 = 2 2 ⟹ x 4 + 4 x 3 + 4 x 2 = 4 (x^2 + 2x)^2 = 2^2 \implies {\color{#3D99F6}x^4 + 4x^3 + 4x^2 = 4}

x 4 + 4 x 3 + 6 x 2 + 4 x − 7 = ( x 4 + 4 x 3 + 4 x 2 ⏟ 4 ) + 2 x 2 + 4 x − 7 = 4 + 2 ( x 2 + 2 x ⏟ 2 ) − 7 = 4 + 2 ( 2 ) − 7 = 1 \begin{aligned}x^4 + 4x^3 + 6x^2 + 4x - 7 & = (\underbrace{\color{#3D99F6}x^4 + 4x^3 + 4x^2}_{4}) + 2x^2 + 4x - 7 \\ & = 4 + 2(\underbrace{\color{#E81990}x^2 + 2x}_{2}) - 7 \\ & = 4 + 2(2) - 7 \\ & = 1 \\ \end{aligned}

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