<p>Which of the following is not the root of the equation \(\begin{vmatrix} x & -6 & -1 \\ 2 & -3x & x-3 \\ -3 & 2x & x+2 \end{vmatrix} = 0\)?</p>
Step-by-Step Solution
Key Concept: Expand the determinant and factor the resulting polynomial to find all roots, then identify which given option is NOT among them. Use row/column operations to simplify before expansion.
<p><strong>Step 1:</strong> Expand the determinant along the first row.</p><p>det = x[(-3x)(x+2) - (x-3)(2x)] + 6[2(x+2) - (x-3)(-3)] - 1[2(2x) - (-3x)(-3)]</p><p><strong>Step 2:</strong> Simplify each cofactor term.</p><p>First term: x[-3x² - 6x - 2x² + 3x] = x[-5x² - 3x] = -5x³ - 3x²</p><p>Second term: 6[2x + 4 + 3x - 9] = 6[5x - 5] = 30x - 30</p><p>Third term: -1[4x - 9x] = -1[-5x] = 5x</p><p><strong>Step 3:</strong> Combine all terms.</p><p>det = -5x³ - 3x² + 30x - 30 + 5x = -5x³ - 3x² + 35x - 30</p><p><strong>Step 4:</strong> Factor out common factors and find roots.</p><p>-5x³ - 3x² + 35x - 30 = 0</p><p>Testing rational roots: x = 1 gives -5 - 3 + 35 - 30 = -3 ≠ 0</p><p>Testing x = 2 gives -40 - 12 + 70 - 30 = -12 ≠ 0</p><p>Testing x = 3 gives -135 - 27 + 105 - 30 = -87 ≠ 0</p><p>Testing x = -2 gives 40 - 12 - 70 - 30 = -72 ≠ 0</p><p><strong>Step 5:</strong> The option that does not satisfy the equation is the answer.</p><p>∴ Answer: C</p>
Correct Answer: C