Mathematics
Algebra, Quadratic Equations, Geometry, Trigonometry, and Statistics
Table of Contents
1. Linear Equations and Inequalities
Linear Equation
An equation where the highest power of the variable is 1. Form: ax + b = c
Solving Linear Equations
Steps:
- Simplify both sides (distribute, combine like terms)
- Move variables to one side, constants to the other
- Isolate the variable by dividing
Example: Solve 3x + 5 = 14
3x = 14 - 5
3x = 9
x = 3
Systems of Linear Equations
Substitution Method
- 1. Solve one equation for one variable
- 2. Substitute into the other equation
- 3. Solve and back-substitute
Elimination Method
- 1. Multiply equations to match coefficients
- 2. Add/subtract to eliminate one variable
- 3. Solve and back-substitute
Linear Inequalities
Solve like equations, BUT: flip the inequality sign when multiplying or dividing by a negative number
<
Less than
>
Greater than
≤
Less than or equal
≥
Greater than or equal
2. Quadratic Equations
Standard Form
ax² + bx + c = 0
where a ≠ 0
Methods of Solving
1. Factoring
Find two numbers that multiply to give c and add to give b
x² + 5x + 6 = 0
(x + 2)(x + 3) = 0
x = -2 or x = -3
2. Quadratic Formula
x = (-b ± √(b² - 4ac)) / 2a
Discriminant (b² - 4ac):
- • D > 0: Two real solutions
- • D = 0: One real solution (repeated root)
- • D < 0: No real solutions (imaginary)
3. Completing the Square
x² + 6x = 7
x² + 6x + 9 = 7 + 9 (add (b/2)² to both sides)
(x + 3)² = 16
x + 3 = ±4
x = 1 or x = -7
Nature of Roots
| Discriminant (D) | Nature of Roots |
|---|---|
| D > 0, perfect square | 2 real, rational, unequal |
| D > 0, not perfect square | 2 real, irrational, unequal |
| D = 0 | 2 real, rational, equal |
| D < 0 | 2 imaginary (not real) |
3. Polynomials
Operations with Polynomials
Addition/Subtraction
Combine like terms (same variable and exponent)
(3x² + 2x) + (x² - 5x) = 4x² - 3x
Multiplication
FOIL or distribute each term
(x + 2)(x + 3) = x² + 5x + 6
Special Products
- (a + b)² = a² + 2ab + b² (Square of binomial sum)
- (a - b)² = a² - 2ab + b² (Square of binomial difference)
- (a + b)(a - b) = a² - b² (Difference of squares)
- (a + b)³ = a³ + 3a²b + 3ab² + b³ (Cube of sum)
Factoring Techniques
Greatest Common Factor (GCF)
6x³ + 9x² = 3x²(2x + 3)
Difference of Squares
x² - 9 = (x + 3)(x - 3)
Trinomial (ax² + bx + c)
x² + 7x + 12 = (x + 3)(x + 4)
4. Geometry
Pythagorean Theorem
a² + b² = c²
For right triangles: c is the hypotenuse (longest side)
Circle Formulas
| Property | Formula |
|---|---|
| Circumference | C = 2πr or C = πd |
| Area | A = πr² |
| Arc Length | L = (θ/360°) × 2πr |
| Sector Area | A = (θ/360°) × πr² |
3D Shapes
| Shape | Surface Area | Volume |
|---|---|---|
| Cube | 6s² | s³ |
| Rectangular Prism | 2(lw + lh + wh) | lwh |
| Cylinder | 2πr² + 2πrh | πr²h |
| Cone | πr² + πrl | ⅓πr²h |
| Sphere | 4πr² | ⁴⁄₃πr³ |
| Pyramid | B + ½pl | ⅓Bh |
5. Trigonometry
SOH-CAH-TOA
Remember the trigonometric ratios!
Trigonometric Ratios
SOH
sin θ = Opposite / Hypotenuse
CAH
cos θ = Adjacent / Hypotenuse
TOA
tan θ = Opposite / Adjacent
Special Angles
| Angle | sin | cos | tan |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | √3/3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | undefined |
Angle of Elevation vs Depression
Angle of Elevation
Looking UP from horizontal line to an object above
Angle of Depression
Looking DOWN from horizontal line to an object below
6. Statistics and Probability
Measures of Central Tendency
Mean (Average)
Mean = Sum / Count
Example: 6 → (2+4+6)/3 = 4
Median (Middle)
Middle value when arranged in order
Example: 9 → Median = 5
Mode (Most Frequent)
Value that appears most often
Example: 4 → Mode = 2
Measures of Dispersion
- Range = Highest - Lowest
- Variance = Average of squared differences from mean
- Standard Deviation = √Variance (measures spread from mean)
Probability
P(Event) = Favorable Outcomes / Total Outcomes
- • P(E) is always between 0 and 1
- • P(certain) = 1
- • P(impossible) = 0
- • P(A') = 1 - P(A) (complement)
Counting Principles
Permutation (Order Matters)
nPr = n! / (n-r)!
Example: Arranging 3 people in 3 seats
Combination (Order Doesn't Matter)
nCr = n! / [r!(n-r)!]
Example: Choosing 3 people from 5
Key Takeaways
- ✓Quadratic Formula: x = (-b ± √(b²-4ac)) / 2a
- ✓Pythagorean Theorem: a² + b² = c²
- ✓SOH-CAH-TOA for trigonometric ratios
- ✓Discriminant: D > 0 (2 real), D = 0 (1 real), D < 0 (no real)
- ✓(a+b)² = a² + 2ab + b²
- ✓Mean = Sum/Count; Median = Middle; Mode = Most frequent
- ✓Volume of Cylinder = πr²h
- ✓P(Event) = Favorable / Total outcomes