If 2x2-x-1-0- then the value of the expression 2x- 1x-4x2-1x22-8x3-1x33 is

2x^2+x 1=0 ⇒ 2x 1x= 1 4x^2+1x^2=(2x 1x)^2+4 =1+4=5 8x^3 1x^3=(2x 1x)^3+3·2x·1x(2x 1x) = 1+6( 1)= 7 ∴2x 1x+(4x^2+1x^2)^2+(8x^3 1x^3)^3 = 1+5^2+( 7)^3= 319 ...

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Byju's Answer

Standard XIII

Mathematics

Roots of a Quadratic Equation

If 2x2+x-1=0,...

Question

If $2 x^{2} + x - 1 = 0$ , then the value of the expression $2 x - \frac{1}{x} + {(4 x^{2} + \frac{1}{x^{2}})}^{2} + {(8 x^{3} - \frac{1}{x^{3}})}^{3}$ is

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Solution

$2 x^{2} + x - 1 = 0$
$\Rightarrow 2 x - \frac{1}{x} = - 1$

$4 x^{2} + \frac{1}{x^{2}} = {(2 x - \frac{1}{x})}^{2} + 4 = 1 + 4 = 5$
$8 x^{3} - \frac{1}{x^{3}} = {(2 x - \frac{1}{x})}^{3} + 3 \cdot 2 x \cdot \frac{1}{x} (2 x - \frac{1}{x}) = - 1 + 6 (- 1) = - 7$

$∴ 2 x - \frac{1}{x} + {(4 x^{2} + \frac{1}{x^{2}})}^{2} + {(8 x^{3} - \frac{1}{x^{3}})}^{3} = - 1 + 5^{2} + (- 7)^{3} = - 319$

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If 2x2+x−1=0, then the value of the expression 2x−1x+(4x2+1x2)2+(8x3−1x3)3 is

2x2+x−1=0 ⇒2x−1x=−1 4x2+1x2=(2x−1x)2+4=1+4=5 8x3−1x3=(2x−1x)3+3⋅2x⋅1x(2x−1x)=−1+6(−1)=−7 ∴2x−1x+(4x2+1x2)2+(8x3−1x3)3=−1+52+(−7)3=−319

If $2 x^{2} + x - 1 = 0$ , then the value of the expression $2 x - \frac{1}{x} + {(4 x^{2} + \frac{1}{x^{2}})}^{2} + {(8 x^{3} - \frac{1}{x^{3}})}^{3}$ is