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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

 Sep 16, 2020
 #1
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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

 Sep 16, 2020
 #2
avatar+732 
+1

wow tysm! that really helped! :)

lokiisnotdead  Sep 16, 2020

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