For what values of $a$ does the equation $9 x^{3} - 2 x + a = 6 - a x$ possess equal roots?

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Solution

The quadratic equation $9 x^{3} - 3 x + a = 6 - a x$ will have equal roots if its discriminant is zero
For equal roots $D = 0$ ,
$\Rightarrow b^{2} - 4 a c = 0$
$\Rightarrow (2 - a)^{2} - 4 \times 9 \times (a - 6) = 0$
$\Rightarrow 4 + a^{2} - 4 a - 36 a + 216 = 0$
$\Rightarrow a^{2} - 40 a + 220 = 0$
By solving the equation by using formula $\frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$ , we get
$x = 33.41, 6.5835$

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