Topics
Linear equations in two variables
- Introduction to linear equations in two variables
- Methods of solving linear equations in two variables
- Simultaneous method
- Simultaneous method
- Substitution Method
- Cross - Multiplication Method
- Graphical Method
- Determinant method
- Determinant of Order Two
- Equations Reducible to a Pair of Linear Equations in Two Variables
- Simple Situational Problems
- Pair of Linear Equations in Two Variables
- Application of simultaneous equations
- Simultaneous method
Quadratic Equations
- Quadratic Equations
- Roots of a Quadratic Equation
- Solutions of Quadratic Equations by Factorization
- Solutions of Quadratic Equations by Completing the Square
- Formula for Solving a Quadratic Equation
- Nature of Roots of a Quadratic Equation
- The Relation Between Roots of the Quadratic Equation and Coefficients
- To Obtain a Quadratic Equation Having Given Roots
- Application of Quadratic Equation
Arithmetic Progression
- Introduction to Sequence
- Terms in a sequence
- Arithmetic Progression
- General Term of an Arithmetic Progression
- Sum of First ‘n’ Terms of an Arithmetic Progressions
- Arithmetic Progressions Examples and Solutions
- Geometric Progression
- General Term of an Geomatric Progression
- Sum of the First 'N' Terms of an Geometric Progression
- Geometric Mean
- Arithmetic Mean - Raw Data
- Concept of Ratio
Financial Planning
Probability
- Probability - A Theoretical Approach
- Basic Ideas of Probability
- Random Experiments
- Outcome
- Equally Likely Outcomes
- Sample Space
- Event and Its Types
- Probability of an Event
- Type of Event - Elementry
- Type of Event - Complementry
- Type of Event - Exclusive
- Type of Event - Exhaustive
- Concept Or Properties of Probability
- Addition Theorem
Statistics
- Tabulation of Data
- Inclusive and Exclusive Type of Tables
- Ogives (Cumulative Frequency Graphs)
- Applications of Ogives in Determination of Median
- Relation Between Measures of Central Tendency
- Introduction to Normal Distribution
- Properties of Normal Distribution
- Concepts of Statistics
- Mean of Grouped Data
- Method of Finding Mean for Grouped Data: Direct Method
- Method of Finding Mean for Grouped Data: Deviation Or Assumed Mean Method
- Method of Finding Mean for Grouped Data: the Step Deviation Method
- Median of Grouped Data
- Mode of Grouped Data
- Concept of Pictograph
- Presentation of Data
- Graphical Representation of Data as Histograms
- Frequency Polygon
- Concept of Pie Graph (Or a Circle-graph)
- Interpretation of Pie Diagram
- Drawing a Pie Graph
Notes
ax2 + bx + c, Divide the polynomial by a ( ∵a ≠ 0) to get `x^2+b/ax+c/a`.
Let us write the polynomial `x^2+b/ax+c/a` in the form of difference of two square numbers. Now we can obtain roots or solutions of equation `x^2+b/ax+c/a` which is equivalent to `ax^2 + bx + c = 0 .`
`ax^2 + bx + c = 0` .............(I)
`x^2+b/ax+c/a=0` ..... dividing both sides by a
`therefore x^2+b/ax+(b/(2a))^2-(b/(2a))^2+c/a=0`
`therefore (x+b/(2a))^2-b^2/(4a^2)+c/a=0`
`therefore(x+b/(2a))^2-(b^2-4ac)/(4a^2)=0` `therefore(x+b/(2a))^2=(b^2-4ac)/(4a^2)`
`therefore(x+b/(2a))=sqrt((b^2-4ac)/(4a^2)) or (x+b/(2a))=-sqrt((b^2-4ac)/(4a^2))`
`therefore x=-b/(2a)+sqrt((b^2-4ac)/(4a^2)) or x=-b/(2a)-sqrt((b^2-4ac)/(4a^2))`
`therefore x=(-b+sqrt(b^2-4ac))/(2a) or x=(-b-sqrt(b^2-4ac))/(2a)`
In short the solution is written as `x=(-b+-sqrt(b^2-4ac))/(2a)` and these values are denoted by `alpha,beta`.
`therefore alpha=(-b+sqrt(b^2-4ac))/(2a) , beta=(-b-sqrt(b^2-4ac))/(2a)` ............................(I)
The values of a, b, c from equation ax2 + bx + c = 0 are substituted in `(-b+-sqrt(b^2-4ac))/(2a)` and further simplified to obtain the roots of the equation. So `x=(-b+-sqrt(b^2-4ac))/(2a)` is the formula used to solve quadratic equation. Out of the two roots any one can be represented by α and the other by β.
That is, instead (I) we can write `alpha=(-b+sqrt(b^2-4ac))/(2a) , beta=(-b-sqrt(b^2-4ac))/(2a)` .......(II)
