If x² - 11x + 1 = 0, what is the value of x⁸ - 14159x⁴ + 11?

Correct option is B

Given:

x^2 - 11x + 1 = 0 

Formula Used: 

$$x+\frac{1}{x} = k$$ 

$$x^2 + \frac{1}{x^2} = k^2 - 2$$​
Solution:

On dividing by  we get,

​$$\Rightarrow x - 11 + \frac{1}{x} = 0\\\ \\\Rightarrow x + \frac{1}{x} = 11\\\ \\\Rightarrow x^2 + \frac{1}{x^2} = (11)^2 - 2\\\ \\x^2 + \frac{1}{x^2} = 119$$​​

Squaring both sides

​$$​​x^4 + \frac{1}{x^4} = 14161 - 2 = 14159 \\\ \\$$​​

Then the value of  x⁸ - 14159x⁴ + 11​​


​$$x^4(x^4 - 14159) + 11\\\ \\x^4(x^4 -x^4- \frac{1}{x^4}) + 11\\\ \\x^4 \left( - \frac{1}{x^4} \right) + 11\\\ \\-1 + 11 = 10​$$​​
x⁸ - 14159x⁴ + 11