Solve for the roots of x in each of the equations below. a. x4 - 81 = 0 b. x4 + 10x2 + 25 = 0 c. x4 - x2 - 6 = 0

a) 0 = x⁴ - 81

   0 = (x² - 9)(x² + 9)

   0 = (x - 3)(x + 3)(x² + 9)  

0 = x - 3  ,  0 = x + 3  ,  0 = x² + 9

  x = 3    ,       x = -3   ,      x² = -9 →  x = √-9  →  x = +/- 3i

real roots 3, 3  ; complex roots 3i, -3i

b) 0 = x⁴ + 10x² + 25

   0 = (x² + 5)²

0 = x² + 5

x² = -5  →  x = +/-√-5  →  x = +/- i√5

no real roots  ; complex roots i√5, -i√5

c) 0 = x⁴ - x² - 6

  0 = (x² - 3)(x² + 2)

0 = x² - 3                       0 = x² + 2

x² = 3  → x = +/-√3        x² = -2  →  x = +/-√-2  →  x = +/-i√2

real roots √3, -√3; complex roots i√2, -i√2

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