Matlab unable to find solution to cubic polynomial

The following cubic equation has three roots.

syms a
solve((1225*a)/2 - 6125 == ((2*a - 35)^2*(60*a + 4200))/840, a)

Matlab's output is:

root(z^3 + 35*z^2 - (8575*z)/2 + 42875, z, 1)
root(z^3 + 35*z^2 - (8575*z)/2 + 42875, z, 2)
root(z^3 + 35*z^2 - (8575*z)/2 + 42875, z, 3)

Can cubic equations like this be solved analytically in Matlab?

Answer 1

[James Tursa]

Tell the solve( ) function that the max degree of the polynomial is 3 to force explicit solutions for the result:

syms a
p = (1225*a)/2 - 6125 - ((2*a - 35)^2*(60*a + 4200))/840
solve(p,a,'MaxDegree',3)

which gives

ans =

28175/(18*(- 5187875/108 + (432^(1/2)*659937359375^(1/2)*1i)/432)^(1/3)) + ((432^(1/2)*659937359375^(1/2)*1i)/432 - 5187875/108)^(1/3) - 35/3

- 28175/(36*(- 5187875/108 + (432^(1/2)*659937359375^(1/2)*1i)/432)^(1/3)) - ((432^(1/2)*659937359375^(1/2)*1i)/432 - 5187875/108)^(1/3)/2 - (3^(1/2)*(28175/(18*(- 5187875/108 + (432^(1/2)*659937359375^(1/2)*1i)/432)^(1/3)) - ((432^(1/2)*659937359375^(1/2)*1i)/432 - 5187875/108)^(1/3))*1i)/2 - 35/3

- 28175/(36*(- 5187875/108 + (432^(1/2)*659937359375^(1/2)*1i)/432)^(1/3)) - ((432^(1/2)*659937359375^(1/2)*1i)/432 - 5187875/108)^(1/3)/2 + (3^(1/2)*(28175/(18*(- 5187875/108 + (432^(1/2)*659937359375^(1/2)*1i)/432)^(1/3)) - ((432^(1/2)*659937359375^(1/2)*1i)/432 - 5187875/108)^(1/3))*1i)/2 - 35/3

Then you can also note

>> simplify(ans)

ans =

(35*2^(1/3)*(- 121 - 1077^(1/2)*3i)^(1/3))/6 + (35*2^(1/3)*(- 121 + 1077^(1/2)*3i)^(1/3))/6 - 35/3

- (35*2^(1/3)*(- 121 + 1077^(1/2)*3i)^(1/3))/12 - (35*2^(1/3)*(1 + 3^(1/2)*1i)*(- 121 - 1077^(1/2)*3i)^(1/3))/12 + (2^(1/3)*3^(1/2)*(- 121 + 1077^(1/2)*3i)^(1/3)*35i)/12 - 35/3

- (35*2^(1/3)*(- 121 + 1077^(1/2)*3i)^(1/3))/12 + (35*2^(1/3)*(- 1 + 3^(1/2)*1i)*(- 121 - 1077^(1/2)*3i)^(1/3))/12 - (2^(1/3)*3^(1/2)*(- 121 + 1077^(1/2)*3i)^(1/3)*35i)/12 - 35/3

>> imag(ans)

ans =

0

0

0

So you can pick off the real part for the answer.