For what values of $a$ is the ratio of the roots of the equation $ax^2 - \left(a + 3\right)x + 3 = 0$ equal to $1.5$?
Given quadratic equation: $\;\;\;$ $ax^2 - \left(a + 3\right)x + 3 = 0$ $\;\;\; \cdots \; (1)$
Comparing equation $(1)$ with the standard quadratic equation $\;$ $Ax^2 + Bx + C = 0$ $\;$ gives
$A = a, \; B = - \left(a + 3\right), \; C = 3$
Roots of equation $(1)$ are
$x = \dfrac{a + 3 \pm \sqrt{\left[-\left(a + 3\right)\right]^2 - 4 \times a \times 3}}{2 \times a}$
i.e. $\;$ $x = \dfrac{a + 3 \pm \sqrt{a^2 + 6a + 9 - 12a}}{2a}$
i.e. $\;$ $x = \dfrac{a + 3 \pm \sqrt{a^2 - 6a + 9}}{2a}$
i.e. $\;$ $x = \dfrac{a + 3 \pm \sqrt{\left(a - 3\right)^2}}{2a}$
i.e. $\;$ $x = \dfrac{a + 3 \pm \left(a - 3\right)}{2a}$
$\therefore \;$ The roots are $\;\;$ $x_1 = \dfrac{a + 3 + a - 3}{2a} = 1$, $\;$ $x_2 = \dfrac{a + 3 - a + 3}{2a} = \dfrac{3}{a}$
As per question, ratio of roots $= 1.5$
i.e. $\;$ $\dfrac{x_1}{x_2} = \dfrac{1}{3/a} = 1.5$ $\implies$ $a = 4.5$
or $\;$ $\dfrac{x_2}{x_1} = 1.5 = \dfrac{3/a}{1} = 1.5$ $\implies$ $a = 2$