| x 5 2 |
|-6 4 1 | = x(squared) + 22x - 1
| 3 1 x |
Solve for x.
To solve for x, we need to find the values that satisfy the equation x² + 22x - 1.
Step 1: Multiply each element in the matrix by its corresponding cofactor.
| x 5 2 | | 4 1 | | -6 1 | | -6 4 | | 3 1 |
|-6 4 1 | = x * | 3 1 | - 5 * | 3 1 | + 2 * | -6 1 | = 6x + 23
| 3 1 x | | -6 4 | | -6 4 | | -6 4 | | -6 4 |
Step 2: Set up the equation using determinants.
x² + 22x - 1 = | x 5 2 |
|-6 4 1 |
| 3 1 x |
Step 3: Calculate the determinant of the matrix.
| x 5 2 |
D = | -6 4 1 |
| 3 1 x |
= (x * (4 * x - 1) - 5 * (-6x - 2) + 2 * (-6 - 4))
= 4x² - x + 30x +10 -12 - 8
= 4x² + 29x - 10
Step 4: Substitute the determinant into the equation.
4x² + 29x - 10 = x² + 22x - 1
Step 5: Bring all terms to one side of the equation to set it equal to zero.
4x² + 7x - 9 = 0
Step 6: Solve the quadratic equation using factoring, completing the square, or the quadratic formula.
We can use the quadratic formula to solve for x:
x = (-b ± √(b² - 4ac)) / 2a
Plugging in the values from the quadratic equation:
a = 4, b = 7, c = -9
x = (-7 ± √(7² - 4 * 4 * -9)) / (2 * 4)
x = (-7 ± √(49 + 144)) / 8
x = (-7 ± √193) / 8
Therefore, the solutions for x are ( -7 + √193 )/ 8 and ( -7 - √193 )/ 8.