For what values of $a$ does the equation $9x^2 - 2x + a = 6 - ax$ possess equal roots?
Given quadratic equation: $\;$ $9x^2 -2x+a = 6 - ax$
i.e. $\;$ $9x^2 + \left(a-2\right)x + a-6 = 0$ $\;\;\; \cdots \; (1)$
Comparing equation $(1)$ with the standard quadratic equation $\;$ $Ax^2 + Bx + C = 0$ $\;$ gives
$A = 9, \; B = a-2, \; C = a-6$
Discriminant of equation $(1)$ is
$\Delta = B^2 - 4AC = \left(a-2\right)^2 - 4 \times 9 \times \left(a - 6\right)$
i.e. $\;$ $\Delta = a^2 - 4a + 4 - 36a + 216$
i.e. $\;$ $\Delta = a^2 - 40a + 220$
For equal roots, $\;$ $\Delta = 0$
i.e. $\;$ $a^2 - 40a + 220 = 0$
i.e. $\;$ $a = \dfrac{40 \pm \sqrt{1600 - 4 \times 1 \times 220}}{2}$
i.e. $\;$ $a = \dfrac{40 \pm \sqrt{1600-880}}{2}$
i.e. $\;$ $a = \dfrac{40 \pm \sqrt{720}}{2} = \dfrac{40 \pm 12 \sqrt{5}}{2} = 20 \pm 6 \sqrt{5}$