Algebra cheat sheet

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.

Useful Links

• Paul's Online Notes
• Khan Academy Algebra
• Purplemath's Algebra Lessons
• MathHelp.com Algebra 1

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^(n-m)
• (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

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 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^(n-1) + ax^(n-2) + ... + a^(n-1))
• x^n + a^n = (x + a)(x^(n-1) - ax^(n-2) + a^2x^(n-3) - ... + a^(n-1))

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

• 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)

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 y-intercept (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.