# Is the quadratic formula a complete formular for solving the quadratic equation? [closed]

If the quadratic formula is complete formula for solving the quadratic equation why is i= [(5±i√7)/2]+i one of the infinitely many solutions of x^2+3x+5=0? If that is so are the formulas for cubic or quartic equations complete? Why for example does -8+3i and 17+2i fit to be roots of the same equation is it a coincidence or is it that the quadratic formula is incomplete? If the quadratic formula is incomplete what is the guarantee that the cubic or even equation formulas are complete? I realize that the roots I sent could not solve the problem. I was investigating a condition in which if: x^2+a_1 x+a_0=0 ---------- 1 If we permit a complex number, x=c+di ---------2 Then 〖(c+di)〗^2+a_1 (c+di)+a_0=0 ----------- 3 Expanding and simplifying: c^2-d^2+a_1 c+a_0+i(2cd+a_1 d)=0 ------------ 4 Equation the real terms of the expansion to zero: c^2-d^2+a_1 c+a_0=0 ---------------- 5 Equating the imaginary terms of the expansion to zero: 2cd+a_1 d=0 ---------------- 6 Adding together 5 and 6: c^2+c(a_1+2d)+a_1 d+a_0=0 ------------- 7 Solving 7: c=(a_1+2d±√(2&((a_1+2d)^2-4(a_1 d+a_0-d^2)))/2 ------------- 8 Substituting 8 into 2: x=(a_1+2d±√(2&((a_1+2d)^2-4(a_1 d+a_0-d^2)))/2+di ------- 9 When d = 0 the quadratic equation is recovered. This was a surprising result to me and I was wondering if equation 9 could be used to produce results using any random number d. I discovered it couldn’t. I realized that the only way for the imaginary coefficient to be zero Is for d to be equal to zero, in which case the quadratic formula as it is formulated is complete.

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## closed as unclear what you're asking by The Chaz 2.0, azimut, Davide Giraudo, Hagen von Eitzen, Daniel FischerOct 16 '13 at 18:26

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What did you actually mean to write between "why is" and "one of the..." ?? –  DonAntonio Oct 16 '13 at 16:38
$(-8 + 3 i)^2 + 3 (-8 + 3 i) + 5 = 36-39 i \neq 0$. None of your other "roots" work either. Why do you think they do? –  Macavity Oct 16 '13 at 17:29