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Lesson 250 min read
Mathematics
Algebra, Geometry, Number Properties & Word Problems
In This Lesson
Algebra Essentials
Quadratic Formula
x = (-b ± √(b² - 4ac)) / 2a
For equation ax² + bx + c = 0
Discriminant (b² - 4ac):
- > 0: Two real solutions
- = 0: One real solution (double root)
- < 0: No real solutions (complex)
Factoring Patterns
- Difference of Squares: a² - b² = (a + b)(a - b)
- Perfect Square Trinomial: a² + 2ab + b² = (a + b)²
- Sum of Cubes: a³ + b³ = (a + b)(a² - ab + b²)
- Difference of Cubes: a³ - b³ = (a - b)(a² + ab + b²)
Exponent Rules
- • aᵐ × aⁿ = aᵐ⁺ⁿ
- • aᵐ ÷ aⁿ = aᵐ⁻ⁿ
- • (aᵐ)ⁿ = aᵐⁿ
- • a⁰ = 1
- • a⁻ⁿ = 1/aⁿ
- • (ab)ⁿ = aⁿbⁿ
Geometry Formulas
| Shape | Area | Perimeter |
|---|---|---|
| Rectangle | l × w | 2(l + w) |
| Triangle | ½ × b × h | a + b + c |
| Circle | πr² | 2πr (circumference) |
| Trapezoid | ½(b₁ + b₂) × h | Sum of all sides |
Pythagorean Theorem
a² + b² = c²
Where c is the hypotenuse (longest side)
Common Pythagorean Triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25
Number Properties
Divisibility Rules
- By 2: Last digit is even (0, 2, 4, 6, 8)
- By 3: Sum of digits is divisible by 3
- By 4: Last two digits form a number divisible by 4
- By 5: Last digit is 0 or 5
- By 6: Divisible by both 2 and 3
- By 9: Sum of digits is divisible by 9
- By 10: Last digit is 0
Prime Numbers (1-50)
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
Remember: 1 is NOT prime, 2 is the only even prime
LCM and GCF
- GCF (Greatest Common Factor): Largest number that divides both numbers evenly
- LCM (Least Common Multiple): Smallest number that both numbers divide into evenly
- Relationship: GCF × LCM = Product of the two numbers
Word Problems
Age Problems
Set up equations using variables for current ages.
- • "X years ago" = subtract X
- • "In X years" = add X
Work Problems
Rate × Time = Work Done
Combined rate = sum of individual rates
Distance Problems
Distance = Rate × Time
- • Same direction: subtract speeds
- • Opposite direction: add speeds
Mixture Problems
Amount₁ × Conc₁ + Amount₂ × Conc₂ = Total × Final Conc
Percentages & Ratios
Percentage Shortcuts
- • 50% = ½ (divide by 2)
- • 25% = ¼ (divide by 4)
- • 20% = ⅕ (divide by 5)
- • 10% = move decimal one place left
- • 1% = move decimal two places left
Percentage Formulas
- Percent Change: [(New - Old) / Old] × 100
- Finding the Whole: Part ÷ Percentage = Whole
- Finding the Part: Whole × Percentage = Part
Mental Math Shortcuts
Multiplication Tricks
- • Multiply by 5: Divide by 2, then ×10
- • Multiply by 9: Multiply by 10, subtract original
- • Multiply by 11: Add digits, put in middle
- • Square numbers ending in 5: n5² = n(n+1) + 25
Quick Estimates
- • Round to nearest 10 or 100
- • Use compatible numbers
- • Break complex problems into parts
- • Check answer reasonableness