If $$x + \frac{1}{x} = \frac{{\sqrt 3 + 1}}{2},$$ then what is the value of $${x^4} + \frac{1}{{{x^4}}}?$$
A. $$\frac{{4\sqrt 3 - 1}}{4}$$
B. $$\frac{{4\sqrt 3 + 1}}{2}$$
C. $$\frac{{ - 4\sqrt 3 - 1}}{4}$$
D. $$\frac{{ - 4\sqrt 3 - 1}}{2}$$
Answer: Option C
Solution(By Examveda Team)
$$\eqalign{ & x + \frac{1}{x} = \frac{{\sqrt 3 + 1}}{2} \cr & {x^2} + \frac{1}{{{x^2}}} = {\left( {\frac{{\sqrt 3 + 1}}{2}} \right)^2} - 2 \cr & = \frac{{4 + 2\sqrt 3 }}{4} - 2 \cr & = \frac{{4 + 2\sqrt 3 - 8}}{4} \cr & = \frac{{ - 4 + 2\sqrt 3 }}{4} \cr & = \frac{{ - 2 + \sqrt 3 }}{2} \cr & {x^4} + \frac{1}{{{x^4}}} = {\left( {\frac{{ - 2 + \sqrt 3 }}{2}} \right)^2} - 2 \cr & = \frac{{4 + 3 - 4\sqrt 3 - 8}}{4} \cr & = \frac{{ - 1 - 4\sqrt 3 }}{4} \cr} $$Related Questions on Algebra
If $$p \times q = p + q + \frac{p}{q}{\text{,}}$$ then the value of 8 × 2 is?
A. 6
B. 10
C. 14
D. 16
A. $$1 + \frac{1}{{x + 4}}$$
B. x + 4
C. $$\frac{1}{x}$$
D. $$\frac{{x + 4}}{x}$$
A. $$\frac{{20}}{{27}}$$
B. $$\frac{{27}}{{20}}$$
C. $$\frac{6}{8}$$
D. $$\frac{8}{6}$$
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