1. Basic Algebra 1.1 Variables and Expressions Example 1: Simplify 2 ( x

5 ) − 3 x 2(x+5)−3x. Explanation: Distribute and combine like terms: 2 x + 10 − 3 x

− x + 10 2x+10−3x=−x+10.

Example 2: Evaluate 3 a − 4 b 3a−4b when a

2 a=2, b

1 b=1. Explanation: Substitute values: 3 ( 2 ) − 4 ( 1 )

6 − 4

2 3(2)−4(1)=6−4=2.

1.2 Linear Equations Example 1: Solve 2 x + 3

7 2x+3=7. Explanation: Subtract 3: 2 x

4 2x=4, then divide by 2: x

2 x=2.

Example 2: Solve 4 x − 5

2 x + 3 4x−5=2x+3. Explanation: Subtract 2 x 2x: 2 x − 5

3 2x−5=3, add 5: 2 x

8 2x=8, divide: x

4 x=4.

1.3 Quadratic Equations Example 1: Solve x 2 − 5 x + 6

0 x 2 −5x+6=0. Explanation: Factor into ( x − 2 ) ( x − 3 )

0 (x−2)(x−3)=0, so x

2 x=2 or 3 3.

Example 2: Solve 2 x 2 + 4 x − 6

0 2x 2 +4x−6=0. Explanation: Use quadratic formula: x

− 4 ± 16 + 48 4

1 x= 4 −4± 16+48 ​

​ =1 or − 3 −3.

1.4 Inequalities Example 1: Solve 3 x − 5 < 10 3x−5<10. Explanation: Add 5: 3 x < 15 3x<15, divide: x < 5 x<5.

Example 2: Solve − 2 ≤ 4 x + 6 < 10 −2≤4x+6<10. Explanation: Subtract 6: − 8 ≤ 4 x < 4 −8≤4x<4, divide: − 2 ≤ x < 1 −2≤x<1.

2. Functions 2.1 Function Basics Example 1: If f ( x ) = 2 x

1 f(x)=2x+1, find f ( 3 ) f(3). Explanation: Substitute: f ( 3 )

2 ( 3 ) + 1

7 f(3)=2(3)+1=7.

Example 2: Find f ( a + h ) f(a+h) for f ( x )

x 2 f(x)=x 2 . Explanation: Substitute: ( a + h ) 2

a 2 + 2 a h + h 2 (a+h) 2 =a 2 +2ah+h 2 .

2.2 Linear Functions Example 1: Graph y

2 x − 1 y=2x−1. Explanation: Slope = 2, y-intercept at ( 0 , − 1 ) (0,−1).

Example 2: Find the slope between ( 1 , 3 ) (1,3) and ( 4 , 9 ) (4,9). Explanation: Slope

9 − 3 4 − 1

2

4−1 9−3 ​ =2.

2.3 Quadratic Functions Example 1: Graph y

x 2 − 4 x + 3 y=x 2 −4x+3. Explanation: Vertex at ( 2 , − 1 ) (2,−1), opens upward.

Example 2: Convert y

2 x 2 + 8 x + 5 y=2x 2 +8x+5 to vertex form. Explanation: Complete the square: y

2 ( x + 2 ) 2 − 3 y=2(x+2) 2 −3.

3. Intermediate Algebra 3.1 Systems of Equations Example 1: Solve 2 x

y

7 2x+y=7, 3 x − y

3 3x−y=3. Explanation: Add equations: 5 x

10 5x=10, so x

2 x=2, then y

3 y=3.

Example 2: Solve y

2 x + 1 y=2x+1, y

− x + 4 y=−x+4. Explanation: Set equal: 2 x + 1

− x + 4 2x+1=−x+4, solve: x

1 x=1, y

3 y=3.

3.2 Polynomials Example 1: Multiply ( x + 2 ) ( x − 3 ) (x+2)(x−3). Explanation: Use FOIL: x 2 − 3 x + 2 x − 6

x 2 − x − 6 x 2 −3x+2x−6=x 2 −x−6.

Example 2: Divide ( 3 x 3 − 2 x 2 + 4 ) ÷ ( x − 1 ) (3x 3 −2x 2 +4)÷(x−1). Explanation: Use synthetic division: Quotient

3 x 2 + x + 1 =3x 2 +x+1, remainder 5 5.

3.3 Factoring Example 1: Factor x 2 + 5 x + 6 x 2 +5x+6. Explanation: Find two numbers multiplying to 6 and adding to 5: ( x + 2 ) ( x + 3 ) (x+2)(x+3).

Example 2: Factor 9 x 2 − 16 9x 2 −16. Explanation: Difference of squares: ( 3 x − 4 ) ( 3 x + 4 ) (3x−4)(3x+4).

4. Advanced Algebra 4.1 Logarithms Example 1: Solve log ⁡ 2 ( x ) = 5 log 2 ​ (x)=5. Explanation: Convert to exponential form: x = 2 5 = 32 x=2 5 =32.

Example 2: Simplify log ⁡ 3 ( 81 ) − log ⁡ 3 ( 9 ) log 3 ​ (81)−log 3 ​ (9). Explanation: Use properties: log ⁡ 3 ( 81 / 9 )

log ⁡ 3 ( 9 )

2 log 3 ​ (81/9)=log 3 ​ (9)=2.

4.2 Complex Numbers Example 1: Simplify ( 3 + 2 i ) + ( 1 − 4 i ) (3+2i)+(1−4i). Explanation: Combine real and imaginary parts: 4 − 2 i 4−2i.

Example 2: Multiply ( 2 + i ) ( 3 − 2 i ) (2+i)(3−2i). Explanation: Use FOIL: 6 − 4 i + 3 i − 2 i 2

8 − i 6−4i+3i−2i 2 =8−i.

4.3 Matrices Example 1: Add [ 2 3 1 4 ] + [ 5 0 − 2 3 ] [ 2 1 ​

3 4 ​ ]+[ 5 −2 ​

0 3 ​ ]. Explanation: Result: [ 7 3 − 1 7 ] [ 7 −1 ​

3 7 ​ ].

Example 2: Multiply [ 1 2 3 4 ] ⋅ [ 5 6 7 8 ] [ 1 3 ​

2 4 ​ ]⋅[ 5 7 ​

6 8 ​ ]. Explanation: Result: [ 19 22 43 50 ] [ 19 43 ​

22 50 ​ ].

5. Pre-Calculus Topics 5.1 Trigonometry Example 1: Find sin ⁡ ( 4 5 ∘ ) sin(45 ∘ ). Explanation: sin ⁡ ( 4 5 ∘ ) = 2 2 sin(45 ∘ )= 2 2 ​

​ .

Example 2: Solve cos ⁡ ( θ )

0.5 cos(θ)=0.5. Explanation: Solutions: θ

6 0 ∘ θ=60 ∘ or 30 0 ∘ 300 ∘ .

5.2 Sequences and Series Example 1: Find the 10th term of 3 , 7 , 11 , 15 , … 3,7,11,15,…. Explanation: Arithmetic sequence: a 10

3 + ( 9 ) ( 4 )

39 a 10 ​ =3+(9)(4)=39.

Example 2: Sum 2 + 4 + 8 + ⋯ + 64 2+4+8+⋯+64. Explanation: Geometric series: S

2 ( 1 − 2 6 ) 1 − 2

126 S= 1−2 2(1−2 6 ) ​ =126.