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Question 3

If $$x^2 + 6x + 1 = 0$$ and $$\frac{x^4+kx^2+1}{3x^3+kx^2+3x} = 2$$, then the value of $$k$$ is

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Dividing $$x^2+6x+1=0$$ by $$x$$ gives $$x+\frac{1}{x}=-6$$, and hence $$x^2+\frac{1}{x^2}=34$$. Dividing the given fraction by $$x^2$$ changes it to $$\frac{k+34}{k-18}=2$$. Therefore $$k+34=2k-36$$, which gives $$k=70$$.

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