Solving complex equation How to solve x ^ 4 + 4 = 0

x^4=-4
X ^ 2 = 2I or - 2I
X = 1 + I or - 1-I or 1-I or - 1 + I
There are four roots, all of which are imaginary roots

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20. Please help me to analyze my doubts in the following problems, exercise: the equation lg2x / LG (x + a) = 2, ask why a is worth, the equation has a solution? arrangement: lg2x=2lg(x+a) 2x=(x+a)^2 The results are as follows x^2+2(a-1)x+a^2=0 And 2x > 0, x + a > 0, For the above univariate quadratic equation, △ = 4 [(A-1) ^ 2] - 4 (a ^ 2) = - 8A + 4, There are three cases ① When △ 0, - 8A + 4 > 0, a < 1 / 2 In this case, the equation has two solutions, X = {2-2a ± [radical (4-8a)} / 2 = 1-A ± [radical (1-2a)] In this case, x = (1-A) + [radical (1-2a)] > 0 is obviously true (positive number plus positive number); For x = (1-A) - [radical (1-2a)], since (1-A) ^ 2 - (1-2a) = 1-2a + A ^ 2-1 + 2A = a ^ 2 > 0, x = 1-A - [radical (1-2a)] > 0 also holds However, because x + a > 0 is required, Therefore, when a ＜ 1 / 2 and X + a ＞ 0, the original equation has two solutions ② When △ = 0, a = 1 / 2 In this case, the equation is x ^ 2-x + 1 / 4 = 0, and the unique solution is x = 1 / 2 But it is meaningless that the denominator is 0 Therefore, when x = 1 / 2, the original equation has no solution ③ When △ 0, a ＞ 1 / 2, the original equation has no solution To sum up, (1) When a < 1 / 2, the equation has two solutions; (2) There is no a such that the equation has a solution; (3) When a ≥ 1 / 2, the equation has no solution My question is: 1, "for x = (1-A) - [root (1-2a)], since (1-A) ^ 2 - (1-2a) = 1-2a + A ^ 2-1 + 2A = a ^ 2 > 0", how can I get 1-A - √ (1-2a) > 0 from this? 2. "When a ＜ 1 / 2 and X + a ＞ 0, the original equation has two solutions." why? From 2x > 0, we can get x > 0; X + a > 0, we can get x > - A. why not find the maximum value of - x, and then combine A0 and 2x > 0, that is, x > 0 and x > - A. should we also find the maximum value of - x, and then find the range of a? Even if we can't find it, should we also meet a > - x? Why only a > 2