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Lesson 155 min read
Mathematics
Algebra, Geometry, Trigonometry, Statistics & Problem Solving
In This Lesson
Algebra
Quadratic Equations
ax² + bx + c = 0
Quadratic Formula: x = (-b ± √(b² - 4ac)) / 2a
Discriminant (D = b² - 4ac):
- D > 0: Two distinct real roots
- D = 0: One repeated real root
- D < 0: Two complex roots (no real solutions)
Systems of Equations
Methods to solve:
- Substitution: Solve one equation for a variable, substitute into other
- Elimination: Add/subtract equations to eliminate one variable
- Graphing: Find intersection point of lines
Polynomial Operations
- FOIL Method: (a+b)(c+d) = ac + ad + bc + bd
- Difference of Squares: a² - b² = (a+b)(a-b)
- Perfect Square: a² + 2ab + b² = (a+b)²
- Sum/Difference of Cubes: a³ ± b³ = (a ± b)(a² ∓ ab + b²)
Exponents & Logarithms
Exponent Rules:
- aᵐ × aⁿ = aᵐ⁺ⁿ
- aᵐ ÷ aⁿ = aᵐ⁻ⁿ
- (aᵐ)ⁿ = aᵐⁿ
Logarithm Rules:
- log(ab) = log a + log b
- log(a/b) = log a - log b
- log(aⁿ) = n × log a
Geometry
| Shape | Area | Perimeter/Circumference |
|---|---|---|
| Rectangle | l × w | 2(l + w) |
| Triangle | ½bh | a + b + c |
| Circle | πr² | 2πr |
| Trapezoid | ½(b₁ + b₂)h | Sum of all sides |
Pythagorean Theorem
a² + b² = c²
Common triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25
Trigonometry
SOH-CAH-TOA
SOH
sin θ = Opposite / Hypotenuse
CAH
cos θ = Adjacent / Hypotenuse
TOA
tan θ = Opposite / Adjacent
Common Angle Values
| Angle | sin | cos | tan |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | undefined |
Statistics & Probability
Measures of Central Tendency
- Mean: Sum ÷ Count
- Median: Middle value when sorted
- Mode: Most frequent value
Probability
- P(A): Favorable outcomes ÷ Total outcomes
- P(A and B): P(A) × P(B) if independent
- P(A or B): P(A) + P(B) - P(A and B)
Number Theory
Divisibility Rules
- By 2: Last digit is even
- By 3: Sum of digits divisible by 3
- By 5: Ends in 0 or 5
- By 9: Sum of digits divisible by 9
- By 10: Ends in 0
Problem-Solving Strategies
Time-Saving Tips
- ✓ Skip hard problems, return later
- ✓ Estimate before calculating
- ✓ Use elimination for multiple choice
- ✓ Check if answer is reasonable
Common Mistakes
- ✗ Sign errors in algebra
- ✗ Forgetting to simplify
- ✗ Misreading the question
- ✗ Unit conversion errors