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So... okay maybe it doesn't immediately sound useful for a quadratic, but as soon as you want to solve cubic (x3) or quartic (x4) equations, you'll discover that the "equivalent" to the quadratic formula for them is maddeningly complicated. Complex analysis becomes a light leading out of the tunnel for this.

Oh I remember having studied this, and had to use the cubic formula, it was insanely complicated. Basically if I remember well solving arbitrary equation of 3rd and 4th order is possible but insanely complicated, and 5th order and upper is basically impossible without resorting to numerical approximations. Or you can be smart and factor the equation away by "guessing" solutions, but it's not always possible.

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That's what I thought. Who the hell says "-5 meters above sea level"?

Well, the sign simply shows the direcition. The same could be said about latitude and longitude positioning, the sign indicate the direction whether it's north/south or east/west. Basically it carries extra information in the numebr itself. Complex numbers is the same concept, they carry information for a number and an angle at the same time.

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It is not the square root of -1, but rather it is the "imaginary" number that when squared will equal -1. i2 = -1

Actually every (positive real) number has two "square roots" (i.e. two reals that multiplied by themselves gives this result) but the "square root" function only returns the positive one. For example both 3 and -3 will give 9 when multiplied with themselves, but only 3 is the square root of 9, not -3.

Similarly, both i and -i gives -1 when multipplied by themselves. Since neither is "positive" nor "negative" (they are something else entierely !) we can't decide which one is the "square root" of -1. We could decide that we keep the one in the first hemicycle (angle is smaller than 180°), this is simple when having the square root of negative numbers (the two "roots" will always be conjugated), but when taking the square root of an already complex number there is no simple solution (the two "roots" can be anything if I'm not mistaken).