Algebra Core Topics Mastery Guide

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Question: Provide a comprehensive guide to the following Algebra topics: 1. Variables and Expressions, 2. Linear Equations (one variable), 3. Solving Linear Inequalities, 4. Graphing Linear Equations (slope-intercept form), 5. Systems of Linear Equations (solving methods), 6. Exponents and Radicals (properties), 7. Polynomials (addition, subtraction, multiplication), 8. Factoring Polynomials (various techniques), 9. Quadratic Equations (standard form), 10. Solving Quadratic Equations (factoring, quadratic formula), 11. Graphing Quadratic Functions (parabolas), 12. Rational Expressions (simplifying, operations), 13. Solving Rational Equations, 14. Functions (definition, notation, domain, range), 15. Types of Functions (linear, quadratic, exponential), 16. Inverse Functions, 17. Logarithmic Functions (definition, properties), 18. Solving Exponential and Logarithmic Equations, 19. Sequences and Series (arithmetic, geometric), 20. Absolute Value Equations and Inequalities.

Answer:

Here is a comprehensive yet concise guide covering high school Algebra 1 and Algebra 2 core topics. Each section includes key definitions, rules, methods, and examples.

1. Variables and Expressions

A variable is a symbol (usually a letter) that represents an unknown value.
A constant is a fixed number.
An algebraic expression combines variables, constants, and operations (no = sign).
Terms are parts separated by + or −; like terms have the same variable(s) and exponents.
Coefficient is the number multiplying the variable.

Example: Simplify 3x + 5 − 2x + 7
→ (3x − 2x) + (5 + 7) = x + 12

2. Linear Equations (one variable)

Goal: Isolate the variable. Use inverse operations and the Golden Rule: Whatever you do to one side, do to the other.

Steps for ax + b = cx + d:

Move variable terms to one side.
Move constants to the other side.
Divide by coefficient of variable.

Example: 3(x − 4) = 5x + 2
3x − 12 = 5x + 2
−12 − 2 = 5x − 3x
−14 = 2x
x = −7

3. Solving Linear Inequalities

Similar to equations, but flip the inequality sign when multiplying or dividing by a negative number.

Rules:

Add/subtract: sign stays the same.
Multiply/divide by positive: sign stays.
Multiply/divide by negative: sign flips.

Example: −2x + 5 > 11
−2x > 6
x < −3 (flipped)

Compound inequalities: −3 ≤ 2x + 1 < 7 → −4 ≤ 2x < 6 → −2 ≤ x < 3

4. Graphing Linear Equations (slope-intercept form)

Slope-intercept form: y = mx + b

m = slope (rise/run, steepness)
b = y-intercept (where line crosses y-axis)

Steps:

Plot b on y-axis.
From there, use slope (rise/run) to find another point.
Draw line through points.

Example: y = −3/2 x + 4
Start at (0, 4), down 3 right 2 → (2, 1), etc.

5. Systems of Linear Equations (solving methods)

Two or more linear equations solved together. Solutions: one point (consistent), no solution (parallel), infinitely many (same line).

Methods:

Graphing: intersection point.
Substitution: solve one for a variable, substitute into other.
Elimination: add/subtract equations to eliminate a variable.

Example (elimination):
2x + 3y = 8
4x − 3y = 10
Add → 6x = 18 → x = 3
Then 2(3) + 3y = 8 → y = 2/3

6. Exponents and Radicals (properties)

Exponent rules:

xᵐ · xⁿ = xᵐ⁺ⁿ
xᵐ / xⁿ = xᵐ⁻ⁿ
(xᵐ)ⁿ = xᵐⁿ
x⁰ = 1 (x ≠ 0)
x⁻ⁿ = 1/xⁿ
(xy)ⁿ = xⁿ yⁿ
(x/y)ⁿ = xⁿ/yⁿ

Radical rules (√ = ^{1/2}):

√(ab) = √a · √b
√(a/b) = √a / √b
√(a²) = a
Rationalize denominator: multiply by conjugate.

7. Polynomials (addition, subtraction, multiplication)

Polynomial: terms with non-negative integer exponents.
Degree: highest exponent.
Addition/Subtraction: combine like terms.
Multiplication: distribute (FOIL for binomials).

Example: (2x² + 3x − 5)(x − 4) = 2x³ − 8x² + 3x² − 12x − 5x + 20 = 2x³ − 5x² − 17x + 20

8. Factoring Polynomials (various techniques)

Greatest Common Factor (GCF)
Difference of Squares: a² − b² = (a − b)(a + b)
Perfect Square Trinomial: a² ± 2ab + b² = (a ± b)²
Grouping
Trinomials (ac method or trial): x² + bx + c → factors of c that add to b
Sum/Difference of Cubes: a³ ± b³ = (a ± b)(a² ∓ ab + b²)

Example: x² − 7x − 18 = (x − 9)(x + 2)

9. Quadratic Equations (standard form)

Standard form: ax² + bx + c = 0 (a ≠ 0)

10. Solving Quadratic Equations

Methods:

Factoring (preferred when possible)
Square root property: x² = k → x = ±√k
Completing the square
Quadratic formula: x = [-b ± √(b² − 4ac)] / (2a)
- Discriminant D = b² − 4ac: D > 0 (two real), D = 0 (one real), D < 0 (no real)

Example: 2x² − 5x − 3 = 0 → x = [5 ± √(25 + 24)] / 4 = [5 ± 7]/4 → x = 3 or x = −1/2

11. Graphing Quadratic Functions (parabolas)

Vertex form: y = a(x − h)² + k → vertex (h, k)

a > 0: opens up (minimum)
a < 0: opens down (maximum)
a > 1: narrower; a < 1: wider

Axis of symmetry: x = h
y-intercept: (0, k + ah²)

12. Rational Expressions (simplifying, operations)

Rational expression: fraction of polynomials

Simplify: factor numerator & denominator, cancel common factors (exclude where denominator = 0)

Operations:

Multiply: factor, cancel, multiply
Divide: multiply by reciprocal
Add/Subtract: common denominator, combine

Example: (x² − 4)/(x² − x − 6) = (x−2)(x+2) / (x−3)(x+2) = (x−2)/(x−3) (x ≠ −2, 3)

13. Solving Rational Equations

Find LCD.
Multiply both sides by LCD (clear denominators).
Solve resulting equation.
Check solutions (exclude values that make original denominators zero).

Example: 3/(x−1) + 2/x = 5 → multiply by x(x−1): 3x + 2(x−1) = 5x(x−1)

14. Functions (definition, notation, domain, range)

Function: each input (x) has exactly one output (y) → vertical line test
Notation: f(x) = 2x + 3 → “f of x”
Domain: all possible x-values
Range: all possible y-values

15. Types of Functions

Linear: f(x) = mx + b → straight line
Quadratic: f(x) = ax² + bx + c → parabola
Exponential: f(x) = a·bˣ (b > 0, b ≠ 1) → growth/decay

16. Inverse Functions

f and f⁻¹ undo each other: f(f⁻¹(x)) = x
To find: swap x and y, solve for y.

Example: f(x) = 3x − 7 → y = 3x − 7 → x = 3y − 7 → y = (x + 7)/3 = f⁻¹(x)

One-to-one functions have inverses (horizontal line test).

17. Logarithmic Functions (definition, properties)

Logarithm: log_b(a) = c means b^c = a
Common logs: log = log₁₀, ln = log_e (natural log)

Properties:

log_b(xy) = log_b x + log_b y
log_b(x/y) = log_b x − log_b y
log_b(x^k) = k log_b x
log_b b = 1, log_b 1 = 0
Change of base: log_b a = ln a / ln b

18. Solving Exponential and Logarithmic Equations

Exponential: isolate base, take log of both sides.
Logarithmic: exponentiate both sides (rewrite as exponential).

Example: 5^x = 20 → x = log₅ 20 = ln20 / ln5
Example: log₂(x + 3) = 4 → x + 3 = 2⁴ = 16 → x = 13

19. Sequences and Series

Sequence: ordered list (a₁, a₂, …)
Arithmetic: common difference d
aₙ = a₁ + (n−1)d
Sum Sₙ = n/2 (a₁ + aₙ) or n/2 [2a₁ + (n−1)d]
Geometric: common ratio r
aₙ = a₁ · r^(n−1)
Sum Sₙ = a₁ (1 − r^n)/(1 − r) (|r| < 1 for infinite sum)

20. Absolute Value Equations and Inequalities

x = k → x = k or x = −k (k ≥ 0)
x < k → −k < x < k
x > k → x < −k or x > k

Example:	2x − 3	= 7 → 2x − 3 = 7 or 2x − 3 = −7 → x = 5 or x = −2
Example:	x + 4	≤ 6 → −6 ≤ x + 4 ≤ 6 → −10 ≤ x ≤ 2

References:

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