the solution of -2i is one of the roots.
it's conjugate of +2i is the other root.
the solution of -5+7i is one of the roots.
it's conjugate of -5-71 is the other root.
that's a total of 4 roots which is what you should have since the degree of the equaton is fourth degree.
remember:
complex roots always come in pairs.
a complex root is a root with a real part and an imaginary part.
the real part of -2i is equal to 0
that root is really 0 - 2i and 0 + 2i
any time you see just an imaginary part, you can be sure that the real part is equal to 0 and the complex root is really 0 +/- the imaginary part.
the real part of 0 is not normally shown.
if we multiply the roots of this equation, you will be able to see the quadratic equation itself.
you will not, however, see the graph crossing the x-axis since the graph will only cross the x-axis where the real roots are.
your equation is found by multiplying the factors together.
the roots of the equation are shown as:
-2i
+2i
-5-7i
-5+7i
the factors of the equation are:
y = (x - 2i) * (x + 2i) * (x - 5 - 7i) * (x - 5 + 7i)
i multiplied those factors out and got the following equation:
y = x^4 - 10x^3 + 78x^2 - 40x + 296
basically, you multiply the complex roots in pairs.
the i part will cancel out.
if i did this right, you should get the following:
(x-2i) * (x+2i) results in x^2 + 4
(x-5-7i) * (x-5+7i) results in x^2 - 10x + 74
you then multiply (x^2 + 4) * (x^2 - 10x + 74) to get x^2 - 10x^3 + 78x^2 - 40x + 296.
the graph of the equation looks like this:
since all the roots of the equation are complex, the equation does not cross the x-axis.
