If (begin{vmatrix} x – 3 & -4 & 3 \ 5 & 2 & 2 \ 2 & -4 & 6 – x end{vmatrix} = -24 ), find the values of x.
Explanation
(begin{vmatrix} x – 3 & -4 & 3 \ 5 & 2 & 2 \ 2 & -4 & 6 – x end{vmatrix} = -24)
((x – 3)[2(6 – x) + 8] + 4[5(6 – x) – 4] + 3[- 20 – 4] = -24)
((x – 3)[12 – 2x + 8] + 4[30 – 5x – 4] + 3[-24] = -24)
((x – 3)(20 – 2x) + 4(26 – 5x) – 72 = -24)
((x – 3)(20 – 2x) + 4(26 – 5x) = – 24 + 72 = 48)
(20x – 2x^{2} – 60 + 6x + 104 – 20x = 48)
(-2x^{2} + 6x – 4 = 0)
(2x^{2} – 6x + 4 = 0)
Solving the equation, we get x = 1 or 2.