How do I solve solve an equation for the first complex root?

I need to be able to solve the following equation for its first root

15.36*L^2 + L*(19592.58957620995934121310710907*i - 1919917.6232407807483900796796661) + (14998852.92303687865065071910875 - 82922.533414158501558648943613169*i)

However, when I use the solve function

solve(15.36*L^2 + L*(19592.58957620995934121310710907*i - 1919917.6232407807483900796796661) + (14998852.92303687865065071910875 - 82922.533414158501558648943613169*i))

Warning: Explicit solution could not be found.

> In solve at 169

ans =

[ empty sym ]

is displayed.

I know the answer is L=7.81235+.0365383i. How can I make MATLAB display this answer?

Answer 1

[Star Strider]

You’re not asking it correctly:

syms L
p = 15.36*L^2 + L*(19592.58957620995934121310710907*i - 1919917.6232407807483900796796661) + (14998852.92303687865065071910875 - 82922.533414158501558648943613169*i);
L = roots(sym2poly(p))

L =

     124.9868e+003 -  1.2756e+003i
       7.8124e+000 + 36.5383e-003i

Answer 2

[Roger Stafford]

This is a quadratic equation of the form "a*x^2+b*x+c=0" with complex-valued coefficients, b and c. The same formula as for reals is valid:

   x = (-b+(+1or-1)*(b^2-4*a*c)^(1/2))/(2*a)

and very likely both roots are complex-valued. Why don't you use that instead of bothering with 'solve'?

The 'sort' function documentation says "The sort function sorts complex elements first by absolute value (magnitude), then by phase angle", so perhaps that is the ordering you had in mind when you said "first root".

Answer 3

[John D'Errico]

You have formulated the problem for solve ENTIRELY correctly, despite what Star said. However, there are often many ways to solve any problem, and a quadratic polynomial is pretty easy any way you do it.

Make sure both i and L are as they should be. Since i is often used as a variable...

syms L
i = sqrt(-1);

res = solve(15.36*L^2 + L*(19592.58957620995934121310710907*i - 1919917.6232407807483900796796661) + (14998852.92303687865065071910875 - 82922.533414158501558648943613169*i))

res =
 206149585070830075/3298534883328 - (25*(23201164600960114908747505178983141/928455029464035206174343168 - (555078919214256089878496645717587*i)/1088033237653166257235558400)^(½))/2 - (67319625179017825*i)/105553116266496

   (25*(23201164600960114908747505178983141/928455029464035206174343168 - (555078919214256089878496645717587*i)/1088033237653166257235558400)^(½))/2 + 206149585070830075/3298534883328 - (67319625179017825*i)/105553116266496

It works fine for me.

vpa(res)
ans =
 7.8123531480905849356308465638994 + 0.036538267898098900099620705189207*i
   124986.82457659024169482817452297 - 1275.5957554690674215826927161319*i

Take your pick of the roots.

[Austin Hazelrig]

What year MATLAB are you running? I have tried this method on both R2010 and R2012. Both gave me an empty sym vector.

[John D'Errico]

The latest/current release (R2014b). It should work on any release that has the symbolic toolbox. Nothing of note has changed that I know of.