Chapter 3 Quadratic Equation

Quadratic Equation: a quadratic equation is any equation having the form ax^2+bx+c=0 where x represents an unknown, and a, b, and c represent known numbers such that a is not equal to 0.

If a = 0, then the equation is linear, not quadratic.

Steps:

     1. Move all equations to the left or right (your choice)
     2. Factor the equation

     3. Put the solution to the original equation.

Example

Solve x² + 5x + 6 = 0

2x² + 9x - 5 = 0

(2x - 1)(x + 5) = 0

2x - 1 =0     OR     x + 5 =0

So, x= 1/2 OR -5

Completing Square

General Formula

The Formula:

Chapter 10 Surface Area and Volume

Surface Area: In general, the surface area is the sum of all the areas of all the shapes that cover the surface of a 3D object.

Volume: Volume is the measure of the amount of space inside of a 3D figure.

Shapes:
     Cube                                        Surface Area: 6 a²

 Volume:  a³

Find the surface area & Volume
 

                 
     Cuboid                                     Surface Area: 2lw + 2lh + 2hw
Volume: l * w * h

Find the surface area and Volume:



     Prism                                      Surface Area: H * ((A * b)*2)
Volume; Base * H

Find the Surface Area & Volume



     Sphere                                        Surface Area: 4  r²  
                Volume:(4/3)  r^{3
Find the surface Area & Volume:}



     Cylinder                                     Surface Area: Area Top and Bottom + area of side
                                                                     (    r ² * 2 )+  Area of side
     Volume:  r² h
Find the surface area and Volume:



     Cone                                          Surface Area:     ( * r * side) + ( r ²⁾

      Volume: 1/3  r² h
Find the surface area and Volume:

Chapter 9 Matrices

Chapter 9 Matrices

 Matrix is a Rectangular arrangement of numbers.

For example:

This matrix has 2 rows (Vertical) and 3 columns (Horizontal). 

Types of matrices
Diagonal matrices: A matrix with non-zero entries only on the diagonal
                               
Upper triangular: A diagonal matrix whose non-zero entries are all 1's
                               
Identity matrices: A matrix with all-zero entries below the top-left-to-lower-right diagonal
                               

There are a number of basic operations that can be applied to matrices, it's called matrix addition, scalar multiplication, transposition, matrix multiplication, row operations, and submatrix.

Addition & Subtracting: to add two matrices, add the numbers in the matching positions.

How its done?
3+4=7                                                8+0=8
4+1=5                                                6+(-9)=-3

How it's done?
3-4=-1                                                8-0=8
4-1=3                                                  6-(-9)=15

In addition and subtracting, the  two matrices must be the same size, meaning the rows and the columns of the matrices should be the same size.

Multiply
   -By a whole number

          How it's done?
               2 * 4=8                       2 * 0=0
               2 * 1=2                       2 * (-9)= -18

   -By another matrices

         How it's done?
                  (1 * 7)+(2 * 9)+(3 * 11)=58 (top left)
                  (1 * 8)+(2 * 10)+(3 * 12)=54 (top right)
                  (4 * 7)+(5 * 9)+(6 * 11)=139 (bottom left)
                  (4 * 8)+(5 * 10)+(6 * 12)=154 (bottom right)

Overall Units

Unit 1: Equation (Chapter 3&5)
Chapter 3 Quadratic Equation
     3:01 Solution using factors
     3:02 Solution by completing the square
     3:03 The quadratic formula
     3:04 Choosing the best method
     3:05 Problems involving quadratic equations
Chapter 5 Further Algebra
     5:01 Simultaneous equations involving a quadratic equation
     5:02 Literal equations: pronumeral restrictions
     5:03 Understanding variables


Unit 2: Curve sketching (Chapter 4, 6&8)
Chapter 4 Number plane graphs
     4:01 The parabola
     4:02 parabola of the form y=ax^2 + bx + c
     4:03 The hyperbola y=k/x
     4:04 Exponential graphs y=a^2
     4:05 The circle
     4:06 Curves of the form y=ax^3 + d

     4:07 Miscellaneous graphs
     4:08 using coordinate geometry to solve problems

Chapter 6 Curve sketching 
     6:01 
     6:02 
     6:03 
     6:04 
     6:05 
Chapter 8 Functions and Logarithms 
     8:01 Functions 
     8:02 Inverse functions 
     8:03 The graph of y=f(x), y=f(x)+k and y=f(x-a)
     8:04 
     8:05 
     8:06 
     8:07 
     8:08

Unit 3: Matrices & Transformation (Chapter 9&14)
Chapter 9 Matrices
     9:01
     9:02
     9:03
     9:04
     9:05
     9:06
Chapter 14 Transformation and matrices
     14:01
     14:02
     14:03
     14:04
     14:05

Unit 4: further geometry (Chapter 10, 11, 13) 
Chapter 10 Surface area and volume 
     10:01 
     10:02 
     10:03 
     10:04 
     10:05 
     10:06 
     10:07 
     10:08 
Chapter 11 Similarity 
     11:01 
     11:02 
     11:03 
     11:04 
     11:05 
     11:06
 Chapter 13 Circle Geometry 
     13:01 
     13:02 
     13:03 
     13:04 
     13:05 
     13:06 
     13:07 
     13:08

Unit 5: Further Trigonometry (Chapter 12) 
Chapter 12 transformations and matrices 
     12:01 
     12:02 
     12:03 
     12:04 
     12:05 
     12:06 
     12:07 
     12:08

Unit 6: Statistic and Probability (Chapter 15&16) 
Chapter 15 Statistics 
     15:01A Review : Representing data 
     15:01B Review : Analysing data 
     15:02 Using the standard deviation 
     15:03 The Normal Distribution 
     15:04 Statistics with two variables 
Chapter 16 Probability 
     16:01 Probability review 
     16:02A Simultaneous events 
     16:02B Successive events 
     16:03 Independent events and conditional probability