The Algebra Cheat Sheet is a compact guide designed to reinforce your algebraic fundamentals, providing quick references to essential formulas, properties, and concepts. It serves as a handy tool for students to review and check their work while studying or doing homework. This cheat sheet aims to boost confidence and efficiency in algebraic problemsolving by making key information readily accessible.
General Tips for Algebra
Understanding the Basics
 Remember the order of operations (PEMDAS/BODMAS).
 Familiarize yourself with basic algebraic properties (commutative, associative, distributive).
 Always check for factors that can be simplified before solving equations.
Working with Equations
 Keep equations balanced by performing the same operation on both sides.
 When dealing with fractions, find a common denominator to combine terms.
 To isolate a variable, reverse operations in the reverse order they were applied.
Factoring
 Look for a greatest common factor (GCF) to simplify expressions.
 Recognize patterns in polynomials that fit special factoring formulas, like difference of squares and perfect square trinomials.
 Factor by grouping when you have four terms.
Functions and Graphs
 Understand the shape and main features of basic graphs (linear, quadratic, etc.).
 Use a table of values if you're unsure about the shape of a graph.
 Remember that the vertical line test can be used to determine if a graph represents a function.
Inequalities
 When multiplying or dividing by a negative number, flip the inequality sign.
 Graph inequalities on a number line or coordinate plane to visualize the solution set.
Exponents and Radicals
 Practice the laws of exponents regularly to become proficient.
 Remember that radical expressions can often be simplified.
 Know the difference between a rational exponent and a radical; they are alternative forms of the same expression.
Complex Numbers
 Use i to represent the square root of 1 and apply it to simplify square roots of negative numbers.
 When multiplying complex numbers, use the FOIL method and remember that i^2 = 1.
Logarithms
 Convert between exponential and logarithmic form to simplify solving problems.
 Use properties of logarithms to combine or break apart logs for easier computation.
Word Problems
 Read the problem carefully and identify variables, constants, and relationships between them.
 Translate the words into algebraic expressions or equations.
 Check your solutions in the context of the problem to ensure they make sense.
Practice and Review
 Regular practice is essential for mastery.
 Review errors to understand and learn from mistakes.
 Use additional resources such as online tutorials, textbooks, and study groups when needed.
Basic Properties and Facts
Arithmetic Operations
 ab + ac = a(b + c)
 (a/c) * (b/c) = ab/c^2
 (a/b) / (c/d) = (a/b) * (d/c) = ad/bc
 (a/b)  (c/d) = (ad  bc)/bd
 a  b / c  d = b  a / d  c
 (a + b) / c = a/c + b/c
 (ab + ac) / a = b + c, a ≠ 0
 (a/b) / (c/d) = ad/bc
Properties of Inequalities
 If a < b then a + c < b + c and a  c < b  c
 If a < b and c > 0 then ac < bc and a/c < b/c
 If a < b and c < 0 then ac > bc and a/c > b/c
Properties of Absolute Value
 a = { a if a ≥ 0; a if a < 0 }
 a ≥ 0
 a = a
 ab = a b
 a/b = a / b
 a + b ≤ a + b (Triangle Inequality)
Exponent Properties
 a^n * a^m = a^(n+m)
 (ab)^n = a^n * b^n
 (a^n)^m = a^(nm)
 a^0 = 1, a ≠ 0
 a^n / a^m = a^(nm)
 (a/b)^n = a^n / b^n
 a^n = 1 / a^n
 (a/b)^n = (b/a)^n
 a^n = 1 / a^n
Properties of Radicals
 sqrt(n)(a^n) = a if n is odd
 sqrt(n)(a^n) = a if n is even
 sqrt(n)(a) * sqrt(n)(b) = sqrt(n)(ab)
 sqrt(n)(a) / sqrt(n)(b) = sqrt(n)(a/b)
 sqrt(m)(sqrt(n)(a)) = sqrt(mn)(a)
 sqrt(n)(a^m) = a^(m/n)
Distance Formula
 If P1 = (x1, y1) and P2 = (x2, y2) are two points the distance between them is d(P1, P2) = sqrt((x2  x1)^2 + (y2  y1)^2)
Complex Numbers
 i = sqrt(1), i^2 = 1
 sqrt(a) = i*sqrt(a), a > 0
 (a + bi) + (c + di) = a + c + (b + d)i
 (a + bi)  (c + di) = a  c + (b  d)i
 (a + bi) * (c + di) = ac  bd + (ad + bc)i
 (a + bi) * (a  bi) = a^2 + b^2
 a + bi = sqrt(a^2 + b^2) (Complex Modulus)
 (a + bi) = a  bi (Complex Conjugate)
 (a + bi) * (a + bi) = a + bi^2
Logarithms and Log Properties
Definition + Example
 y = log_b(x) is equivalent to x = b^y
 log_5(125) = 3 because 5^3 = 125
Special Logarithms
 ln(x) = log_e(x) natural log
 log(x) = log_10(x) common log where e = 2.718281828...
Logarithm Properties
 log_b(b) = 1
 log_b(1) = 0
 b^(log_b(x)) = x
 log_b(x^r) = r log_b(x)
 log_b(xy) = log_b(x) + log_b(y)
 log_b(x/y) = log_b(x)  log_b(y)
 The domain of log_b(x) is x > 0
Factoring and Solving
Factoring Formulas
 x^2  a^2 = (x + a)(x  a)
 x^2 + 2ax + a^2 = (x + a)^2
 x^2  2ax + a^2 = (x  a)^2
 x^2 + (a + b)x + ab = (x + a)(x + b)
 x^3 + 3ax^2 + 3a^2x + a^3 = (x + a)^3
 x^3  3ax^2 + 3a^2x  a^3 = (x  a)^3
 x^3 + a^3 = (x + a)(x^2  ax + a^2)
 x^3  a^3 = (x  a)(x^2 + ax + a^2)
 x^n  a^n = (x^n  a^n)(x^n + a^n)
 If n is odd then, x^n  a^n = (x  a)(x^(n1) + ax^(n2) + ... + a^(n1))
 x^n + a^n = (x + a)(x^(n1)  ax^(n2) + a^2x^(n3)  ... + a^(n1))
Quadratic Formula
 Solve ax^2 + bx + c = 0, a ≠ 0
 x = (b ± √(b^2  4ac)) / (2a)
 If b^2  4ac > 0  Two real unequal solns.
 If b^2  4ac = 0  Repeated real solution.
 If b^2  4ac < 0  Two complex solutions.
Square Root Property
 If x^2 = p then x = ±√p
Absolute Value Equations/Inequalities
 If b is a positive number p = b => p = b or p = b p < b => b < p < b p > b => p < b or p > b
Completing the Square
Completing
 Solve 2x^2  6x  10 = 0 (1) Divide by the coefficient of the x^2 x^2  3x  5 = 0 (2) Move the constant to the other side x^2  3x = 5 (3) Take half the coefficient of x, square it and add it to both sides x^2  3x + (3/2)^2 = 5 + (3/2)^2 x^2  3x + 9/4 = 29/4 (4) Factor the left side (x  3/2)^2 = 29/4 (5) Use Square Root Property x  3/2 = ±√(29/4) (6) Solve for x x = 3/2 ± √(29/4)
Functions and Graphs
Constant Function
 y = a or f(x) = a
 Graph is a horizontal line passing through the point (0, a).
Line/Linear Function
 y = mx + b or f(x) = mx + b
 Graph is a line with point (0, b) and slope m.
Slope
 Slope of the line containing the two points (x1, y1) and (x2, y2) is m = (y2  y1) / (x2  x1)
Slope – intercept form
 The equation of the line with slope m and yintercept (0, b) is y = mx + b
Point – Slope form
 The equation of the line with slope m and passing through the point (x1, y1) is y = y1 + m(x  x1)
Parabola/Quadratic Function

y = a(x  h)^2 + k or f(x) = a(x  h)^2 + k

The graph is a parabola that opens up if a > 0 or down if a < 0 and has a vertex at (h, k).

y = ax^2 + bx + c or f(x) = ax^2 + bx + c

The graph is a parabola that opens up if a > 0 or down if a < 0 and has a vertex at (b / 2a, b / 2a).
Circle
 (x  h)^2 + (y  k)^2 = r^2
 Graph is a circle with radius r and center (h, k).
Ellipse
 (x  h)^2 / a^2 + (y  k)^2 / b^2 = 1
 Graph is an ellipse with center (h, k) with vertices a units right/left from the center and vertices b units up/down from the center.
Hyperbola

(x  h)^2 / a^2  (y  k)^2 / b^2 = 1

Graph is a hyperbola that opens left and right, has a center at (h, k), vertices a units left/right of center and asymptotes that pass through center with slope ±b / a.

(y  k)^2 / b^2  (x  h)^2 / a^2 = 1

Graph is a hyperbola that opens up and down, has a center at (h, k), vertices b units up/down from the center and asymptotes that pass through center with slope ±a / b.