Quadratic substitution question: applying substitution $p=x+\frac1x$ to $2x^4+x^3-6x^2+x+2=0$

Question

By using the substitution $p=x+\frac{1}{x}$, show that the equation $$2x^4+x^3-6x^2+x+2=0$$ reduces to $2p^2+p-10=0$.

I can't think of anything that produces a useful result, I tried writing p as $p=\frac{x^2+1}{x}$ and finding areas to substitute but have come with no progress. Could someone offer a slight hint on how to proceed?

Thanks

Answer 1

Like this,

as the equation is Reciprocal Equation of the First type with $x\ne0,$

divide either sides by $\displaystyle x^{\frac42}=x^2$ to reduce the degree of the equation by half

$$2x^2+x-6+\frac1x+\frac2{x^2}=0$$

$$\implies 2\left(x^2+\frac1{x^2}\right)+\left(x+\frac1x\right)-6=0$$

$$\implies 2\left\{\left(x+\frac1x\right)^2-2\right\}+\left(x+\frac1x\right)-6=0$$

Reference : Reciprocal Equation is explained here:

Chapter XI of Higher Algebra,Barnard & Child and

Article $568−570$ of Higher algebra, Hall & Knight

Answer 2

Hint: Try working backwards. Start with: $$ 2\left(x + \dfrac{1}{x}\right)^2 + \left(x + \dfrac{1}{x}\right) - 10 = 0 $$ then try to obtain the original equation by expanding then clearing the denominators.

Answer 3

$$p=x+x^{-1}$$ $$p^2=x^2+2+x^{-2}$$ $$2x^4+x^3-6x^2+x+2=0$$ $$2x^2+x-6+x^{-1}+2x^{-2}=0$$ $$2(x^2+2+x^{-2})+(x+x^{-1})-10=0$$ $$2p^2+p-10=0$$

Answer 4

$$ \begin{align} 2x^4+x^3-6x^2+x+2 & = x^2\Big(2x^2 + x - 6 + \frac1x + \frac{2}{x^2}\Big) \\[12pt] & = x^2 \Big(2\left(x+\frac1x\right)^2 + \left(x+\frac1x\right) - 10 \Big) \\[12pt] & = x^2(2p^2 +p-10). \end{align} $$ This equals $0$ only if $x=0$ or the second factor equals $0$. But clearly $x=0$ is not one of the solutions.