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If $\frac{x}{y}$ + $\frac{y}{x}$ = 3 Find $\frac{x^2}{y^2}$ + $\frac{y^2}{x^2}$

Any Ideas on how to begin ?

user360471
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    Hint try to solve the equation $t+t^{-1} = 3$ where $t = y/x$ – Zau Aug 21 '16 at 03:32
  • A general problem-solving strategy: what could you do to the given to get what you want? Here, the obvious thing to try is squaring, since doing so will generate something that more closely resembles the expression you're after. Sometimes the obvious thing to try doesn't work, but one can always hope. – symplectomorphic Aug 21 '16 at 03:57

3 Answers3

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\begin{align*} \frac{x}{y} + \frac{y}{x} = 3 &\Longrightarrow \left(\frac{x}{y} + \frac{y}{x}\right)^2 = 9 \\ &\Longrightarrow \frac{x^2}{y^2} + \frac{y^2}{x^2} = 9-2=7 \end{align*}

S. Bryant
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  • Why was 2 subtracted from nine? – user360471 Aug 21 '16 at 06:13
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    Since $\displaystyle 2\cdot \frac{x}{y} \cdot \frac{y}{x} = 2$. – S. Bryant Aug 21 '16 at 07:36
  • I don't get it still.Can you explain a little bit more – user360471 Aug 21 '16 at 08:25
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    It is just the use of perfect square formula. Here is the thing: \begin{align} &\quad\quad\left( \frac{x}{y} + \frac{y}{x} \right)^2 = 9 \ &\Longrightarrow \left(\frac{x}{y}\right)^2 + 2 \cdot \frac{x}{y} \cdot \frac{y}{x} + \left( \frac{y}{x} \right)^2 = 9 \ &\Longrightarrow \frac{x^2}{y^2} + \frac{y^2}{x^2} = 9-2=7. \end{align} – S. Bryant Aug 21 '16 at 08:34
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Start by squaring both sides of $\frac{x}{y}+\frac{y}{x}=3$. What happens to the cross terms?

carmichael561
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Hint: Look at:

$$\left(\frac{x}{y} + \frac{y}{x}\right)^2=3^2=9$$

Why to look at this? We know if we expand we will have a $(\frac{x}{y})^2=\frac{x^2}{y^2}$ term and a $(\frac{y}{x})^2=\frac{y^2}{x^2}$ term so this might be worth a try.

Ahmed S. Attaalla
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