#!/usr/bin/env python
# coding: utf-8

# # Parasitic solutions

# In[25]:


from sympy import *
init_printing()


# Find $x$ such that
# $$
# \exp(-x) = 99 x
# $$

# In[26]:


x = symbols('x')
eq1 = exp(-x) - 99*x
r1 = solve(eq1,x)
for r in r1:
    print(N(r))


# Let us make a Taylor expansion of $\exp(-x)$ around $x=0$

# In[27]:


series(exp(-x),x,0)


# We truncate this at the quadratic term and consider the equation
# $$
# 1 - x + \frac{x^2}{2} = 99 x \qquad \Longrightarrow \qquad x^2 - 200 x + 2 = 0
# $$

# In[28]:


eq2 = x**2 - 200*x + 2
r2 = solve(eq2,x)
for r in r2:
    print(N(r))


# The simplied problem has more solutions than the original problem. One of the solutions is the correct one while the other is a spurious or parasitic solution.