+738 2 solutions of \(x^4 - 3x^3 + 5x^2 - 27x - 36 = 0\) are pure imaginary numbers. Find these 2 solutions.
I honestly have no idea how to start since this equation is so complex. Thank you in advance to anyone who helps! I really, really appreciate it! :D
Solve for x:
x^4 - 3 x^3 + 5 x^2 - 27 x - 36 = 0
The left hand side factors into a product with three terms:
(x - 4) (x + 1) (x^2 + 9) = 0
Split into three equations:
x - 4 = 0 or x + 1 = 0 or x^2 + 9 = 0
Add 4 to both sides:
x = 4 or x + 1 = 0 or x^2 + 9 = 0
Subtract 1 from both sides:
x = 4 or x = -1 or x^2 + 9 = 0
Subtract 9 from both sides:
x = 4 or x = -1 or x^2 = -9
Take the square root of both sides:
x = 4 or x = -1 or x = 3 i or x = -3 i
