The sum of the real roots of the equation | cases x & - 6 & - 1 2 &...

Given \left|\begin{cases}x & - 6 & - 1 \\ 2 & - 3x & x - 3 \\ - 3 & 2x & x + 2\end{cases}\right| = 0

On expansion, we get $x\left\{- 3x\left(x + 2\right) - 2x\left(x - 3\right)\right\} + 6\left\{2\left(x + 2\right) + 3\left(x - 3\right)\right\} - 1\left(4x - 9x\right) = 0$

$⇒ x\left\{- 3{x}^{2} - 6x - 2{x}^{2} + 6x\right\} + 6\left\{2x + 4 + 3x - 9\right\} - 1\left(- 5x\right) = 0$

$⇒ - 5{x}^{3} + 30x - 30 + 5x = 0$

$⇒ 5{x}^{3} - 35x + 30 = 0$

$⇒ {x}^{3} - 7x + 6 = 0$

$⇒ \left(x - 1\right)\left(x - 2\right)\left(x + 3\right)$

⇒ x = 1 or $x = 2 or x = - 3 .</p><p>Here all roots \left\{1, 2, - 3\right\} are real, hence the sum of the real roots is$1 + 2 - 3 = 0 .