Efficient way to solve this (nonlinear) equation?

Hello all, I want to solve a nonlinear equation but don't know efficient way to do it in Matlab. Can anyone suggest a good way? Here is my code:

G12=7000; 
phi=2*pi/180;

syms gamma_6
tau_6= G12*(gamma_6-20*(gamma_6)^2);

sigma_x= -tau_6/(phi+gamma_6);
sigma_1=vpa(diff(sigma_x,gamma_6));

f=matlabFunction(sigma_1);

I want to solve expression sigma_1=0; sigma_1 is function of gamma_6. What I can do is solve till

sigma_1=vpa(diff(sigma_x,gamma_6));

Then copy entire string to solve('.....=0',gamma_6) However this is not most efficient way. I tried fsolve(f,..) but problems with algorithm. Any suggestions most appreciated. Thanks!

Answer 1

[Walter Roberson]

-(1/90)*pi +/- (1/180)*sqrt(4*pi^2 + 18*pi)

sigma_1 has two parts added together. When you bring them to a common denominator the expression simplifies to

((25200000 * gamma_6 - 630000)*pi + 1134000000*gamma_6^2) / (pi+90*gamma_6)^2

You want to solve for 0 so you can solve for the numerator being 0. That is a simple quadratic.