What are all the solutions of the equation?

Question

X⁶ = 81X²   and the imaginary solutions

Answer 1

x⁶ = 81x²

 

x⁶ - 81x² = 0

 

x²(x⁴ - 81) = 0

 

x²(x² - 9)(x² + 9) = 0

 

x²(x - 3)(x + 3)(x² + 9) = 0

 

Setting each factor equal to 0, we get x = 0, 3, -3, 3i, -3i

Answer 2

First we can divide by x^2 to get the equation x^4=81...and note the real roots as +/- 3.

 

We next note that i=\sqrt{-1} has the property that i^4=1 and (-i)^4=1, so +/-3i are also solutions to this quartic equation.

 

In conclusion we have four complex roots, 3, -3, 3i, and -3i.

 

Hope that helps!