Re: Mathematical simplicity

Mathematical simplicity.
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Mathematics isn't always simple.
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But sometimes, with a little thought, we can find very simple shortcuts,
which are, however, surprisingly true.
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We said that in quadratic functions f(x), for example, it was enough to
change the sign of the monomial with an even exponent to obtain the
point-symmetric function $ (for those who follow) called g(x).
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This function g(x) has real roots if f(x) doesn't, having only two
complex roots.
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Now, what does [-b±sqrt(b²-4ac)]/2a become if we change the sign of a?
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It becomes [-b±sqrt(b²+4ac)]/(-2a)
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It's mathematical.
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But that's not all, you'll remember, if you remember anything from what
I said earlier: I said that the complex roots of a function are pure
imaginaries, and that they are found by rotating f(x) 180° about the
point $(0,y₀ ) to form g(x).
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We then have here, directly, the complex roots of f(x), given in pure
imaginaries, as should always be the case if we correctly understand
what we are doing. Notations like x'=2+3i or x"=-3-i are a mathematical
joke.
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Real roots of quadratic functions: x= [-b±sqrt(b²-4ac)]/2a
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Complex roots of quadratic functions:
x= {-[b±sqrt(b²+4ac)]/(2a)}.i
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Beware of sign errors (the big trap of complex roots).
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Example, let's set f(x)=x²+4x+5
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x'=i
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x"=-5i
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If you replace x with i or -5i, you will get f(x)=0.
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If you can't do this, it's because you haven't understood how imaginary
numbers work, like 100% of the human beings on this earth.
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Thank you for your attention.
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R.H.