Solve the following equations:\(\begin{vmatrix}\text{x} & -6 & -1 \\[0

1.	Solve the following equations:\(\begin{vmatrix}\text{x} & -6 & -1 \\[0.3em]2 & -3\text{x} & \text {x} - 3 \\[0.3em]-3 &2\text{x} & \text{x}+ 2 \end{vmatrix}\) = 0\|(x, -6, -1)(2, -3x, x-3)(-3, 2x, x + 2)\| = 0
Answer» Expanding with R₁ 0= x(-3x²- 6x - 2x²+ 6x) + 6(2x+4+3x-9) -1(4x - 9x) 0=x (-5x²) + 6(5x - 5) -1(-5x) 0= -5x³+ 30x - 30 + 5x 0= -5x³+ 35x - 30 x³- 7x + 6 = 0 x³- x - 6x + 6 = 0 x(x²-1) - 6(x - 1) = 0 x(x-1)(x+1) - 6(x - 1) = 0 (x - 1)(x²+ x - 6) = 0 (x-1)(x²+ 3x - 2x - 6) = 0 (x-1)(x(x + 3) - 2(x + 3) =0 (x-1)(x+3)(x-2) = 0 Either x - 1 = 0 or x + 3 = 0 or x - 2 = 0 ∴ x = 1 or x = -3 or x = 2

Solve the following equations:\(\begin{vmatrix}\text{x} & -6 & -1 \\[0.3em]2 & -3\text{x} & \text {x} - 3 \\[0.3em]-3 &2\text{x} & \text{x}+ 2 \end{vmatrix}\) = 0|(x, -6, -1)(2, -3x, x-3)(-3, 2x, x + 2)| = 0

Answer»

Expanding with R₁

0= x(-3x²- 6x - 2x²+ 6x) + 6(2x+4+3x-9) -1(4x - 9x)

0=x (-5x²) + 6(5x - 5) -1(-5x)

0= -5x³+ 30x - 30 + 5x

0= -5x³+ 35x - 30

x³- 7x + 6 = 0

x³- x - 6x + 6 = 0

x(x²-1) - 6(x - 1) = 0

x(x-1)(x+1) - 6(x - 1) = 0

(x - 1)(x²+ x - 6) = 0

(x-1)(x²+ 3x - 2x - 6) = 0

(x-1)(x(x + 3) - 2(x + 3) =0

(x-1)(x+3)(x-2) = 0

Either x - 1 = 0 or x + 3 = 0 or x - 2 = 0

∴ x = 1 or x = -3 or x = 2