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

lokiisnotdead Sep 16, 2020

#1**+1 **

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**

Guest Sep 16, 2020