For the purposes of this paper, I’ll differentiate between real and complex constants and variables by using different fonts for each:

(You need the font Arial Alternative to see the letters below correctly)

 

 

 

Further, as Hardy does in the quote below, I’ll also use lowercase letters, , etc. to refer to the coefficient of the real portion of the associated complex constant and variable, , etc., and uppercase letters, , etc. to refer to the coefficient of the imaginary portion of a complex constant and variable, i.e.  This way you can tell on immediately on sight that a real constant or variable is the real or imaginary coefficient of the associated complex constant or variable.

 

I’ll be investigating conditions for the nature of the roots of the quadratic equation with complex coefficients,

 

 

 

I can only recall seeing one account of investigating conditions for the nature of the roots of the quadratic equation with complex coefficients.  This account is in G.H. Hardy’s book A Course of Pure Mathematics[1], pp. 94-95:[2]

 

14.    The general quadratic equation with complex coefficients.

          This equation is  

         

          Unless  and  are both zero we can divide through by .  Hence         we may consider  as the standard form of our equation.  Putting , and equating real and imaginary parts, we obtain a pair of simultaneous equations for  and , viz.

 

 

If we put

 

 

 

these equations become  

          Squaring and adding we obtain

 

We must choose the signs so that  has the sign of : i.e. if  is positive we must take like signs, if  is negative unlike signs.

 

          Conditions for equal roots.  The two roots can only be equal if both the square roots above vanish, i.e. if  or if   These two conditions are equivalent to the single condition  which expresses the fact that the left-hand side of  is a perfect square.

 

          Conditions for a real root.  If  where  is real, then   Eliminating  we find the required condition is  

 

          Conditions for a purely imaginary root.  This is easily found to be  

 

          Conditions for a pair of conjugate complex roots.  Since the sum and product of two conjugate complex numbers are both real,  and  must both be real, i.e.   Thus the equation  can have a pair of conjugate complex roots only if its coefficients are real.  The reader should verify this conclusion by means of the explicit expressions of the roots.  Moreover, if  the roots will be real even in this case.  Hence for a pair of conjugate complex roots we must have  

 

 

If we put

 

 

 

these equations become  

 

    

We must choose the signs so that  has the sign of : i.e. if  is positive we must take like signs, if  is negative unlike signs.

 

I’ve always felt that Hardy’s substitutions, rearranging, squaring and adding, are somewhat less than intuitive and difficult to follow.

 

I’m also concerned that some of the conditions Hardy’s provides don’t appear to entirely make sense to me.  For instance, Hardy’s condition for a real root,   Since all the terms here contain imaginary coefficients, this condition will always give us zero for a quadratic with all real coefficients.  Obviously, however, quadratics with real coefficients can have complex conjugate roots.

 

Hardy’s condition for purely imaginary roots , , also has me confused.  This implies that a quadratic with all real coefficients will necessarily have purely imaginary roots if .  However, it appears the condition for a quadratic with real coefficients should be .  Again, I don’t understand what Hardy’s talking about.

 

However, my single biggest criticism of Hardy’s account here is his forcing the leading coefficient,  monic, which will completely mask conditions on the coefficients of the general quadratic equation with complex coefficients allowing all coefficients to truly be arbitrary complex numbers.  In other words, Hardy really isn’t treating the general quadratic equation with complex coefficients at all in this account.

 

For instance, I’ll show that a necessary condition for real roots is that  must all be collinear.  Well, that conclusion is masked if  is forced monic, and hence real.

 

You’ll also notice that Hardy uses this form for the quadratic equation:

 

 

whose roots are given by:

 

 

 

This form of the quadratic equation is commonly seen in math books from Hardy’s times and is also still frequently employed, in a sense, in the study of quadratic rings and fields being the trace of  is  and the norm of  is , so that the minimal polynomial for  is , where  is forced monic, so that the roots are simply given by:

 

 

 

which would seem to make sense. (sorry for the divergence here…)

 

Anyway, I don’t see that use of  as one’s general equation, more common in newer math books, or  as one’s general equation, more common in older math books, makes any significant difference.

 

However, I do feel forcing  monic makes a huge difference, and I’m a little disappointed that Hardy choose to analyze the roots of the general quadratic equation with complex coefficients with  forced monic.

 

I struggled with this problem for some time, off and on, trying to understand Hardy’s work better… and not making much headway… and trying to find some other way,… an easier way, at least to my mind, than Hardy’s approach, which also hopefully could also avoid forcing  monic.

 

I believe I’ve successfully accomplished the latter goal, i.e. of avoiding forcing  monic, and also greatly diminished the difficulty in understanding the derivation of the conditions for various natures of the roots.

 

Returning to this equations once again,

 

 

 

my basic strategy is to attempt to use eq’s  and  to establish necessary conditions on the coefficients of , such that it might be possible to express  in terms of , then feed these expressions into  and see if this will yield sufficient conditions, in combination with the previous necessary conditions established with  eq’s  and  on the coefficients of , such that conditions similar to Hardy’s can be established for various natures of the roots with arbitrary complex coefficients.

 

Unless I’ve made some mistake(s), I believe I’ve successfully accomplished this end, except in the case of equal roots, where I’ve yet to find a (new!) way in which the symmetric functions are helpful.

 

First of all, let’s reexpress eq’s  and  using only real numbers:

 

 

 

Next, we simplify eq’s  and  by separating the real and imaginary portions:

 

Conditions for real roots.  It’s clear from  that it’s necessary for  resp.  Since a complex number can be seen as defining a line with respect to the origin,

 

 

 

which also implies  since parallelism on the Euclidean plane is a transitive relation.  Further, since all of these lines share a common point, i.e. the origin, they are, in fact, collinear.

 

Since we’ve established that it’s necessary for  to be collinear to have real roots, it’s possible to express  as real scalar multiples of  so that our general equation then becomes:

 (I’ll demonstrate the validity of the derivation of the real scalar multiplies,  shortly)

 

and hence,

 

 

 

From this it’s evident that we’ll have real roots if  or , which upon substituting back  gives us something quite familiar:

 

 

 

So, geometrically, it’s evident that to have real

roots we need to have the cross products for the “vectors” for  equal to zero to indicate co linearity, i.e. , AND  squared must be greater than or equal to 4 times the product of .

 follows since:

 

 

 

which is the condition upon the real components of  stating that , and further that  are, in fact, collinear since they share the origin as a common point.

 

 can be reexpressed to give another form of the  condition for the co linearity of  in terms of the ratios between the real and imaginary parts of :

 

 

 

Now, bearing in mind the immediately proceeding coefficient ratio derivations, i.e. (11a), (11b), of two parallel, collinear in this case, lines or vectors, I’ll demonstrate the validity of the derivation of the real scalar multiples, :

 

So, real roots are determined if

 

 

 

AND

 

 

 

Conditions for a pair of complex conjugate roots.  This case is closely related to the case for real roots.

 

Since the sum and product of two complex conjugate numbers is a real number, it’s evident, as with the case for real roots, that that it’s necessary for  resp.

 

Just as with arguments above from real roots, this implies that  and further that  are collinear since they share the origin as a common point.

 

Hence, for complex conjugate roots, it is also necessary to have  collinear.

 

And all this leads to the same reexpression of  as real scalar multiples of :

 

 

 

and hence,

 

 

 

However, at this point, the two analyses diverge.

 

From this it’s evident that we’ll have complex conjugate roots if  or .  Which upon substituting back the original coefficient values, we get .

 

So, geometrically, it’s evident that to have complex conjugate roots we need to have the “vectors” for  collinear, i.e. , AND   squared must be less than 4 times .

 

It seems pointless simply to repeat largely equivalent arguments here for the complex conjugate case since they’re virtually identical to those for the real case except for the reversal of the inequality sign.

 

So, the conditions for complex conjugate roots is in terms of the real components of :

 

 and  

 

Conditions for a pair of purely imaginary roots.  This case is a significantly variant from the previous two.

 

First of all, recall these equations:

  

 

It follows from  that it’s necessary for , resp.  Again, since a complex number can be seen as a line with respect to the origin,

 

 

 

So, this means that .

 

We can use the same real scalar multiple for defining  in terms of , but defining  in terms of  will now require incorporating a  or  rotation, which obviously implies multiplication by .

 

As if turns out, the scalar multiplier needed follows precisely from taking the ratio between  just as we did when we needed them to be parallel, as in the two previous cases, i.e.:

 

 

 

However, if we wish  to remain a ‘real’ scalar multiple we’ll need to take the  to the left-hand side:

 

 

 

And this leads to a slightly modified reexpression of  as real scalar multiples of :

 

 

 

 

and hence,

 

 

 

From this it’s evident that we’ll have purely imaginary roots if .  Notice also that if , a form of the roots reflected about the .

 

Transforming back :

 

 

So, geometrically, it’s evident that to have purely imaginary roots we need , i.e.,  perpendicular to  and   and  collinear, i.e. , AND   squared must be greater than or equal to negative 4 times .

 

So, for ease of computation, the conditions for purely imaginary roots is in terms of the real components of :

 and  

 

-         I suspect someone’s going to notice that equations like  have purely imaginary solutions.  If  is zero, it may be meaningless to state that  is perpendicular or parallel to another “vector”.  Although one could also argue that since the dot and cross products of  with another non-zero “vector” will be zero,… so, perhaps the zero “vector” really should be seen as being both perpendicular and parallel to all “vectors”.  For the time being, at least, that’s the convention I’m going to adopt here… that is, that the zero “vector” is both perpendicular and parallel to all non-zero vectors.

-         I believe you’ll find all the formulas will still work as expected with the convention that the zero “vector” is both perpendicular and parallel to all non-zero “vectors”

 

 

Conditions for a pair of equal roots.  This case is a substantially variant from the others.

 

As I stated above, I haven’t been able to find some helpful (new!) way to use the symmetric functions in this case.  So, the most obvious plan of attack is to state that it’s evident that the only way one can have equal roots is if the quantity under the radical in the quadratic formula, i.e. .

 

So, splitting this discriminant into its real and imaginary portions:

 

                    

 

So, the conditions for equal roots is:

 

 

 

Actually, more precisely speaking,  is the “discriminant” of the general univariate quadratic equation.  The discriminant is the product of the squared differences of all 2-pair combinations of all the roots.  For instance, with the quadratic equation the discriminant is .

 

 is an expression of the discriminant in terms of the coefficients…

 

 

 

 

 

 

 

 

Notes:  I did notice that it appears that if there’re equal roots, then this double root is equal to the zero of the first derivative of the quadratic.  However, I haven’t figured out any way to use this to any advantage.

 

 

I haven’t done the bibliography yet.