Question

y⁴+1/y⁴=A

•if A=m⁴+4m²+2 then prove that y²-1=my.​

Answer 1

$\Large\text{\underline{\underline{Based conditions}}}$

I'm regarding the variables restricted to positive numbers in this answer.

Interval of the variables: -

$\cdots\longrightarrow\boxed{m>0,\ y>1.}$

$\Large\text{\underline{\underline{Explanation}}}$

We have a system equation, which is -

$\text{ \cdots\longrightarrow\begin{cases} & y^{4}+\dfrac{1}{y^{4}}=A. \\ & A=m^{4}+4m^{2}+2. \end{cases} }$

If we add $2$ on the second equation, -

$\cdots\longrightarrow A+2=(m^{2}+2)^{2}.$

On the first equation, if we add 2 on both sides, -

$\text{ \cdots\longrightarrow(y^{2}+\dfrac{1}{y^{2}})^{2}=A+2. }$

By substitution on the RHS, -

$\text{ \cdots\longrightarrow(y^{2}+\dfrac{1}{y^{2}})^{2}=(m^{2}+2)^{2}. }$

Now, -

$\text{ \cdots\longrightarrow y^{2}+\dfrac{1}{y^{2}}=m^{2}+2. }$

Let us subtract 2 on both sides, -

$\cdots\longrightarrow (y-\dfrac{1}{y})^{2}=m^{2}.$

Hence, -

$\cdots\longrightarrow y-\dfrac{1}{y}=m.$

Further simplifying gives -

$\text{ \cdots\longrightarrow \boxed{y^{2}-1=my.} }$