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?
$15(2x^2 - y^2) = 7xy$
$\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, $$x:y = 5:6 \; OR \; 3:5$$