# 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.

-

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

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question.If this question can be reworded to fit the rules in the help center, please edit the question.

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