Essential Mathematics Formulas and Concepts

I. Algebra

1. Quadratic Equations

• Standard form: f(x) = ax² + bx + c
• Vertex form: f(x) = a(x - h)² + k (Vertex = (h, k))
• Example: f(x) = x² - 5x + 6 = 0 → (x - 2)(x - 3) = 0 → x = 2 or 3

2. Solving Equations

• Example: (2x/3) + 1 = 5/6
  Subtract 1: 2x/3 = -1/6
  Multiply by 3: 2x = -1/2
  Divide: x = -1/4

3. Function Composition

• (f ∘ g)(x) = f(g(x))
• Example: f(x) = x² + 2x, g(x) = 3x - 1
  (f ∘ g)(2) = f(5) = 25 + 10 = 35

4. Absolute Value

• General form: f(x) = a|x - h| + k
  a = vertical stretch/shrink, h = horizontal shift, k = vertical shift
• Example: g(x) = 2|x - 3| - 1 (Stretch by 2, right 3, down 1)

II. Geometry

5. Parallel Lines & Angles

• Alternate interior angles are equal.
• Example: 5x = 70 → x = 14

6. Pythagorean Theorem

• a² + b² = c²
• Example: leg = 8, hypotenuse = 17 → x² + 64 = 289 → x = 15

7. Similar Triangles

• Use scale factor for sides and perimeters.
• Example: 3:5 ratio, small perimeter = 18 → 18 × (5/3) = 30

8. Midsegment Theorem

• Midsegment = 1/2 × third side.
• Example: DE = 5 → BC = 10

9. Transformations

• Example: y = (x + 3)² - 2 (Left 3, Down 2)

III. Trigonometry

10. SOHCAHTOA

• sin(θ) = opposite / hypotenuse
• cos(θ) = adjacent / hypotenuse
• tan(θ) = opposite / adjacent
• Example: sin(θ) = 4/5 → adjacent = 3 → cos(θ) = 3/5

11. Ladder Problem

• Opposite = 10 × sin(60°) ≈ 10 × 0.866 = 8.7 m

IV. Probability & Statistics

12. Union of Sets

• P(A or B) = P(A) + P(B) - P(A and B)
• Example: 0.3 + 0.4 - 0.1 = 0.6

13. Venn Diagrams

• n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
• Example: 50 + 40 - 10 = 80
• Neither: Total - A - B + Both (Example: 30 - 18 - 15 + 7 = 4)

14. Probability

• Die roll: P(even or >4) = {2, 4, 5, 6} → 4/6 = 2/3
• Without replacement: 5 green out of 12, then 4/11 → (5/12) × (4/11) = 20/132 = 10/33

15. Two-Way Tables

• 60% play sports (120 students); 40% of them are female → 0.6 × 120 = 72 male athletes.

16. Mean (Average)

• Mean = (Sum of values) / (Number of values)
• Example: 10 = (8 + 9 + 11 + 13 + x)/5 → x = 9

V. Extra Formulas

Absolute Value Equations

• |a| = b → a = b or a = -b
• Example: |2x - 6| = 10 → 2x - 6 = 10 or -10 → x = 8 or 2

Tips

• Graphs: Identify vertex, x-intercepts, and y-intercepts.
• Units: Check units (m, %, people) in word problems.
• Fractions: Always simplify.
• Diagrams: Label all components clearly.

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