What kind of equation is z^4-3z^2+2=0 ?

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The equation you have given is z^4 - 3z^2 + 2 = 0. Here you can replace z^2 = x to get a quadratic equation in x which can be solved.

z^4 - 3z^2 + 2 = 0

=> x^2 - 3x + 2 = 0

=> x^2 - 2x - x + 2 = 0

=> x(x - 2) - 1(x - 2) = 0

=> (x - 1)(x - 2) = 0

So we have x = 1 and x = 2

As x = z^2

z^2 = 1 and z^2 = 2

z = -1 and z = +1 and z = +sqrt 2 and - sqrt 2

The roots of the equation are -1 , 1 , sqrt 2 and -sqrt 2

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