Question 1207484: Solve each equation in the complex number system.
A. x^4 = 1
B. x^4 + 3x^2 - 4 = 0
Answer by math_tutor2020(3816)   (Show Source):
You can put this solution on YOUR website!
Part A
x^4 = 1
x^4 - 1 = 0
(x^2)^2 - 1^2 = 0
(x^2-1)(x^2+1) = 0 .... difference of squares rule
(x-1)(x+1)(x^2+1) = 0 .... difference of squares rule used again
x-1 = 0 or x+1 = 0 or x^2+1 = 0
x-1 = 0 solves to x = 1
x+1 = 0 solves to x = -1
x^2+1 = 0 solves to x = i and x = -i where i = sqrt(-1) is an imaginary number.
Answer: The four roots are
1, -1, i, -i
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Part B
Let w = x^2
This will mean w^2 = (x^2)^2 = x^4
x^4 + 3x^2 - 4 = 0
w^2 + 3w - 4 = 0
(w+4)(w-1) = 0 .......... quadratic formula is an alternative to factoring
w+4 = 0 or w-1 = 0
w = -4 or w = 1
If w = -4, then,
w = x^2
x^2 = -4
x = sqrt(-4) or x = -sqrt(-4)
x = 2i or x = -2i
If w = 1, then,
w = x^2
x^2 = 1
x = sqrt(1) or x = -sqrt(1)
x = 1 or x = -1
Answer: The four roots are
1, -1, 2i, -2i
You can use various online calculators (eg: WolframAlpha) to confirm the answers are correct.
If you want to use GeoGebra, then you must be in CAS mode. The command to use is called CSolve. Using the solve command will not produce the complex roots.
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