Symbolic Computation with SymPy

Use SymPy to check algebra, solve equations, inspect domains, and explore responsibly.

Learning objectives

After this lesson you should be able to create SymPy symbols, distinguish expressions from equations, use common symbolic commands, solve and check equations, inspect restrictions, make tables and plots, and compare SymPy with browser symbolic tools.

Motivation: a calculator that understands symbols

A numeric calculator gives decimal values. A symbolic system keeps exact expressions and manipulates formulas. SymPy is a Python computer algebra system. It helps you check and explore, but you still decide the domain and meaning.

Creating symbols and expressions

import sympy as sp

x = sp.symbols("x")
expr = (x + 1)**2
expr

\(\displaystyle \left(x + 1\right)^{2}\)

Use * for multiplication and ** for powers. Write 2*x, not 2x; write x**2, not x^2.

Expressions versus equations

expr = 2*x + 3
equation = sp.Eq(2*x + 3, 11)
expr, equation
(2*x + 3, Eq(2*x + 3, 11))

In Python, = assigns a name. A mathematical equation uses sp.Eq(left, right).

Core commands

Task SymPy command Meaning
Expand sp.expand(expr) multiply out
Factor sp.factor(expr) write as product
Simplify sp.simplify(expr) try a simpler equivalent form
Cancel sp.cancel(expr) simplify rational expression
Together sp.together(expr) combine rational terms 
expr = (x + 2)*(x - 3)
sp.expand(expr), sp.factor(x**2 - x - 6)
(x**2 - x - 6, (x - 3)*(x + 2))
rational_expr = (x**2 - 1)/(x - 1)
sp.cancel(rational_expr)

\(\displaystyle x + 1\)

The result x + 1 needs a warning: the original expression excludes x = 1.

Solving and checking

sp.solve(sp.Eq(3*x - 7, 11), x)
[6]
sp.solve(x**2 - 5*x + 6, x)
[2, 3]
sp.checksol(sp.Eq(3*x - 7, 11), x, 6)
True

For radical equations, checking prevents extraneous answers.

radical_equation = sp.Eq(sp.sqrt(x + 5), x - 1)
[(candidate, sp.checksol(radical_equation, x, candidate)) for candidate in sp.solve(radical_equation, x)]
[(4, True)]

Domains and restrictions

Use sp.denom to inspect rational restrictions:

expr = (x**2 - 9)/(x**2 - x - 6)
denominator = sp.denom(expr)
sp.factor(denominator), sp.solve(sp.Eq(denominator, 0), x)
((x - 3)*(x + 2), [-2, 3])

For logs, require positive inputs. For even roots, require nonnegative radicands. Assumptions can help, but beginners should first write restrictions in words.

Exact and approximate answers

sp.sqrt(2), sp.Rational(1, 3), sp.N(sp.sqrt(2))
(sqrt(2), 1/3, 1.41421356237310)

Exact values are usually better during algebra. Decimals are useful for final interpretation.

Tables and plots

from edumath.algebra import expression_table, function_scene

expression_table("x**2 - 3*x + 2", inputs=(-1, 0, 1, 2, 3, 4))
((-1.0, 6.0), (0.0, 2.0), (1.0, 0.0), (2.0, 0.0), (3.0, 2.0), (4.0, 6.0))
scene = function_scene("x**2 - 3*x + 2", x_min=-2, x_max=5, y_min=-2, y_max=8)
fig, ax = scene.render()

Browser symbolic tools

  • math.js: browser evaluation and symbolic simplification.
  • Nerdamer: browser factoring and solving demos.
  • Algebrite: JavaScript/TypeScript CAS exploration.
  • PyScript: actual Python/SymPy in the browser, with heavier load time.

A PyScript sandbox could load SymPy using a package configuration. Verify the exact current PyScript syntax before publishing a live sandbox.

Practice exercises

  1. Create a SymPy symbol named t.
  2. Write the SymPy expression for (x + 3)^2.
  3. Expand (x + 3)^2.
  4. Factor x^2 - 7x + 12.
  5. Solve 4x - 9 = 15.
  6. Check whether x = 4 solves x^2 - 7x + 12 = 0.
  7. Find the denominator of (x + 1)/(x - 5).
  8. Explain why canceling (x^2 - 1)/(x - 1) needs a warning.
  1. t = sp.symbols("t").
  2. (x + 3)**2.
  3. x**2 + 6*x + 9.
  4. (x - 4)*(x - 3).
  5. x = 6.
  6. Yes, substitution gives 0.
  7. x - 5.
  8. The original expression excludes x = 1.

Subtopic guided practice and checkpoints

Symbolic computation is powerful when you know what question you are asking. Guess the command, run it, and then interpret the result mathematically.

Lab 1: create symbols and parse expressions

Guess first. Should x be typed as a Python string or a SymPy symbol before building x^2 + 1?

Guided exercise.

import sympy as sp
x = sp.Symbol("x")
expr = x**2 + 1

The expression is symbolic, so SymPy can expand, factor, differentiate later, and substitute exact values.

Checkpoint. Create symbols a and b, then build the expression (a + b)^2. Which SymPy command expands it?

Lab 2: expressions versus equations

Guess first. Is x**2 - 4 an equation by itself?

Guided exercise.

expr = x**2 - 4          # expression
Eq(expr, 0)              # equation

An expression can be simplified or factored. An equation can be solved.

Checkpoint. Write a SymPy equation for 3x + 2 = 11. Then solve it.

Lab 3: solve and check

Guess first. Why should you check a symbolic answer after solving?

Guided exercise.

solve gives candidates
checksol tests candidates in the original equation

This matters especially for rational and radical equations, where restrictions or squaring can create invalid candidates.

Checkpoint. Solve sqrt(x + 5) = x - 1 with SymPy, then check each solution in the original equation.

Lab 4: exact versus approximate output

Guess first. Which is more informative: sqrt(2) or 1.41421356?

Guided exercise.

Exact forms preserve structure. Decimal approximations help with measurement and graphing. Use exact answers while solving, then approximate when interpreting.

Checkpoint. Compute an exact value for log(5) and then a decimal approximation. State when each form is useful.

Guessing game checkpoint

Using this lesson with edumath and SymPy

The solver layer expects SymPy objects. If a student starts with typed text, parse it first, then solve the parsed equation.

from edumath.core import parse_equation
from edumath.solvers import solve_equation_steps

parsed = parse_equation("3x + 2 = 11")
solution = solve_equation_steps(parsed)
print(solution.render_text())
Answer: x = 3

Method: linear equation

Steps:

1. Original equation:
Eq(3*x + 2, 11)

2. Move everything to one side:
Eq(3*x - 9, 0)
This writes the equation in the form ax + b = 0.

3. Move the constant term:
Eq(3*x, 9)

4. Divide by the coefficient of the variable:
Eq(x, 3)

Check:

x = 3: valid
import sympy as sp

x = sp.symbols("x")
expr = (x + 2)*(x - 3)
sp.expand(expr), sp.factor(sp.expand(expr))
(x**2 - x - 6, (x - 3)*(x + 2))
equation = sp.Eq(sp.sqrt(x + 5), x - 1)
candidates = sp.solve(equation, x)
[(candidate, sp.checksol(equation, x, candidate)) for candidate in candidates]
[(4, True)]
from edumath.algebra import expand_question

question = expand_question((x + 1)**2)
question.check("x**2 + 2*x + 1").message
'Correct.'

Further reading