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 problem-solving 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^(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
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^(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 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 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
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y = a(x - h)^2 + k or f(x) = a(x - h)^2 + k
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The graph is a parabola that opens up if a > 0 or down if a < 0 and has a vertex at (h, k).
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y = ax^2 + bx + c or f(x) = ax^2 + bx + c
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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
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(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1
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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.
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(y - k)^2 / b^2 - (x - h)^2 / a^2 = 1
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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.