Question

If $$10x^2 - 23xy + 9y^2 = 0$$, find $$x : y$$.

Answer 1

Factor the quadratic equation ($$10 \times 9 = 90$$ and $$-18 - 5 = -23$$):

$$10x^2 - 18xy - 5xy + 9y^2 = 0$$

$$2x(5x - 9y) - y(5x - 9y) = 0$$

$$(5x - 9y)(2x - y) = 0$$ 

Setting each factor to zero:

$$5x - 9y = 0$$

$$\implies 5x = 9y$$

$$\implies \frac{x}{y} = \frac{9}{5}$$

$$2x - y = 0$$

$$\implies 2x = y$$

$$\implies \frac{x}{y} = \frac{1}{2}$$

Therefore, $$x : y = 9 : 5$$ or $$1 : 2$$.