Question

4. Find the ratio e : f, if 5e2 – 13ef – 6f2 = 0 where e and f ≠ 0.

Answer 1

$$5e^2-13ef-6f^2=0$$

Divide both sides by $f^2\not=0$

$$\dfrac{5e^2}{f^2}-\dfrac{13ef}{f^2}-\dfrac{6f^2}{f^2}=0$$

$$5(\dfrac{e}{f})^2-13(\dfrac{e}{f})-6=0$$

Solve the quadratic equation for $\dfrac{e}{f}$

$$D=(-13)^2-4(5)(-6)=289$$

$$\dfrac{e}{f}=\dfrac{13\pm\sqrt{289}}{2(5)}$$

$$\dfrac{e}{f}=\dfrac{13-17}{10}=-0.4$$

or

$$\dfrac{e}{f}=\dfrac{13+17}{10}=3$$

$e: d=-2:5$ or $e:f=3:1$