import sympy as sym
from IPython.display import display, Math

# 1)
q = sym.symbols('q')

eq = 3*q+4/q+3 -5*q-1/q-1
display(Math(sym.latex(eq)))
display(Math(sym.latex(eq.simplify())))
display(Math(sym.latex(sym.cancel(eq))))
display(Math('q=' + sym.latex(sym.solve(eq.simplify()))))

$\displaystyle - 2 q + 2 + \frac{3}{q}$

$\displaystyle - 2 q + 2 + \frac{3}{q}$

$\displaystyle - \frac{1}{q} \left(2 q^{2} - 2 q - 3\right)$

$\displaystyle q=\left [ \frac{1}{2} + \frac{\sqrt{7}}{2}, \quad - \frac{\sqrt{7}}{2} + \frac{1}{2}\right ]$

# 2)
eq = 2*q + 3*q**2 - 5/q - 4/q**3

display(Math(sym.latex(eq)))
display(Math(sym.latex(sym.simplify(eq))))
display(Math(sym.latex(sym.cancel(eq))))

$\displaystyle 3 q^{2} + 2 q - \frac{5}{q} - \frac{4}{q^{3}}$

$\displaystyle 3 q^{2} + 2 q - \frac{5}{q} - \frac{4}{q^{3}}$

$\displaystyle \frac{1}{q^{3}} \left(3 q^{5} + 2 q^{4} - 5 q^{2} - 4\right)$

# 3)
expr = (sym.sqrt(3)+sym.sqrt(15)*q) / (sym.sqrt(2) + sym.sqrt(10)*q)

display(Math(sym.latex(expr)))
display(Math(sym.latex(sym.simplify(expr))))
display(Math(sym.latex(sym.cancel(expr))))

$\displaystyle \frac{\sqrt{15} q + \sqrt{3}}{\sqrt{10} q + \sqrt{2}}$

$\displaystyle \frac{\sqrt{6}}{2}$

$\displaystyle \frac{\sqrt{6}}{2}$

expr.subs(q,10)

(sqrt(3) + 10*sqrt(15))/(sqrt(2) + 10*sqrt(10))

expr.subs(q,10).evalf()

1.22474487139159