Algebra Cheat Sheet

Arithmetic and Fraction Rules

Rule	Statement
Distribute a factor
Multiply a fraction by a number
Fraction divided by a number
Number divided by a fraction
Adding unlike fractions
Subtracting unlike fractions
Dividing two fractions
Canceling a true common factor	,

Exponent Rules

Rule	Statement
Product Rule
Quotient Rule
Power Rule
Power of a Product
Power of a Quotient
Zero Exponent	,
Negative Exponent
Negative Exponent of a Quotient
Rational Exponent

Radical Rules

Rule	Statement
Root as a fractional exponent
Product Rule
Quotient Rule
Root of a root
Odd-index root of a power	( odd)
Even-index root of a power	( even)

Inequalities and Absolute Value

Rule	Statement
Add or subtract the same value	If , then and
Multiply or divide by a positive	If and , then
Multiply or divide by a negative	If and , then — direction flips
Absolute value, by definition	if ; if
Product and quotient	,
Triangle inequality
Solving ()	or
Solving ()
Solving ()	or

Complex Numbers

Rule	Statement
Imaginary unit	,
Square root of a negative	,
Add / subtract	Combine real parts and imaginary parts separately
Multiply	FOIL, then replace every with
Conjugate	The conjugate of is
Conjugate product
Modulus
Divide	Multiply the numerator and denominator by the denominator’s conjugate

Factoring Formulas

Pattern	Factors as
Difference of squares
Perfect square (sum)
Perfect square (difference)
General trinomial
Sum of cubes
Difference of cubes
Difference of even powers

Solving Quadratics

Method	Statement
Quadratic Formula	For ():
Discriminant	Two distinct real solutions
Discriminant	One repeated real solution
Discriminant	Two complex solutions
Square Root Property	If , then

Completing the square, worked example. Solve .

1. Divide every term by the leading coefficient :
2. Move the constant to the other side:
3. Halve the -coefficient, square it, add to both sides:
4. Factor the left side:
5. Apply the Square Root Property:
6. Solve for : , so or

Logarithm Rules

Rule	Statement
Definition	means
Worked example	, because
Identity values	,
Inverse of an exponential	,
Power Rule
Product Rule
Quotient Rule
Natural and common log	;
Domain	is only defined for

Lines and Function Forms

Form	Equation
Slope of a line through two points
Slope-intercept form	, -intercept
Point-slope form
Parabola, vertex form	, vertex
Parabola, standard form	, vertex at
Distance between two points

Conic Sections

Shape	Equation
Circle	, center , radius
Ellipse
Hyperbola (opens left/right)
Hyperbola (opens up/down)

Common Algebra Mistakes

What goes wrong	Why, with an example
treated as or as	Division by zero is undefined — it isn’t any number at all.
	Without parentheses, the exponent binds to only: , while .
	A power of a power multiplies exponents: , not .
	Check with numbers: , but — not equal.
	Only cancels: ; the -term keeps its denominator.
	The negative sign has to distribute to both terms: .
	Expanding fully: — the middle term is real.
	, not — a square root doesn’t distribute over a sum.
	Squaring comes first: , while .
	Any nonzero base to the zero power is , not .