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I was doing some problems out a book I had at home and found this problem:

If $15(2x^2-y^2) = 7xy$, find the ratio of $x : y$

Maybe I've been staring at it for too long (or maybe I'm not thinking correctly), but how would I quickly solve this problem?

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    $\begingroup$ Let the unknown ratio be $\frac{x}{y} = \lambda\,$. Substitute $x = \lambda y$ back into the equation. Solve for $\lambda$. $\endgroup$ – dxiv Jan 1 '18 at 0:44
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$15(2x^2 - y^2) = 7xy$

$\mapsto 30x^2 - 15y^2 - 7xy = 0 $

Let $x:y = r$ $$\mapsto x = ry$$

$30(ry)^2 - 15y^2 - 7(ry)y =0 $ $\mapsto 30r^2y^2 - 15y^2 - 7ry^2 = 0$

Dividing equation by $y^2$:

$30r^2 -15 - 7r = 0 $ $30r^2 -7r - 15 = 0 $

$$Using \ quadratic \ formula$$ $r = \frac{7 \pm \sqrt{7^2 - 4(30)(-15)}}{2(30)}$ $$ r = \frac{5}{6} \; OR \; \frac{3}{5}$$

Therefore, $$x:y = 5:6 \; OR \; 3:5$$

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  • $\begingroup$ Thank you. I appreciate the explanation. $\endgroup$ – ssharma Jan 1 '18 at 18:11
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Hint: divide your equation by $y^2$.

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