LCM And HCF Undersatnding

                                                            LCM and HCF

LCM stands for Lest common multiple.

                    Least ==  smallest                                              Highest = Gratest
                   Common = among number                                  Common = among numbers
                    Multiple =  explained below                                Factors= Divisor

there are two words one is factor and other is multiple.Consider this example 6

we can write 6 like this  3 x 2 = 6

 3 and 2 are factors of 6. It means that 6 is divisible by 2 and 3.

While 6 is multiple of 3 and 2.It means that 6 comes in the table of 3 and 2 so we can get 6 if we multiply 3 and 2

Prime Numbers

Prime Numbers are the numbers with no factors except one and the number itself. for. e.g. 1,2,3,5, 7, 11 and so on. Prime numbers are divisible by 1 and itself .
 Prime number is not divisible by any other numbers. list of Prime numbers 1.

Roman Number System and conversion of European into Roman System

Number system is the mathematical way of representing the things, items by symbols.

For e.g.  In European Number system we represent the numbers of apple in following way

   

Roman numeral is the earliest number system that is still in use. 

                                                                                                                                  

How to Solve Quadratic Equation

  How to Solve Quadratic Equation

What is the meaning of solving Quadratic Equation?

Answer is quite simple "In sample  manner finding the roots of equation"

Now what are  roots of quadratic Equations.

Roots are those values of "x" at which the equation is satisfied for e.g.

                     2x² -8x + 5 = 0   

roots of this equations means those values of x at which this equation is satisfied 

let x1 and x2 be those values of x 

then when we put these values in equation it must satisfy it 

                                     L.H.S=R.H.S 

 

                                   Solving Equation

First we have to find the nature of roots.For this we have to find discriminant

Roots can be real,equal and complex

Following Equation is the genric quadratic equation

                             ax²+bx + c = 0   

                             Discriminant or D is given by

                                D =   b²- 4ac

                If

                             D > 0 two  roots are real

                             D = 0  roots are equal

                             D < 0 two roots are complex 

For e.g.

Case 1:

                                      for the equation

                                        2x² -8x + 5 = 0  

                          Compare it with generic quadratic equation   ax²+bx + c = 0  

                                 a= 2 ,b= -8,c= 5 

                              so D =  (-8)² -4*2*5

                                       = 64 -40

                                       =24

                                      24 >0

                    s        o it has two real roots 

Case 2:

                           Now consider this equation

                               x² -2x + 5 = 0  

                                so a= 1,b= -2,c = 5

                               D=   (-2)² -4*1*5

                                   = 4 -20

                                   = -16

                                 Since -16 <0 font="" nbsp="">

                             so D is negative it has two complex roots 

 

Case 3 :

        

                               x² -2x + 1 = 0  

                              a=1,b=-2,c=1

                            D   =   (-2)² -4*1*1

                                 =  4-4

                                   =0

                           Since D=0 so it has two equal roots

 

                  

         

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Understanding Quadratic Equations

  Iam explaining  quadratic equation in most simple way

"Quadratic Equation" consist of two terms Quadratic and Equation
Dictionary meaning of Quadratic means "Of, relating to, or containing quantities of the second degree."

So Quadratic means term containing power of two

Equations means  " The act or process of equating or of being equated"

Quadratic Equations means "An equation in which one or more of the terms is squared but raised to no higher power, having the general form ax² + bx + c = 0, where a, b, and c are constants."

Quad stand for 2 any equation which have highest power of 2 is quadratic equation
For e.g.

                            5x² + 6x + 8= 0

                           2x² -8x + 5 = 0   

Now consider this example

5x² + 5x +15

Is it a quadratic equation
Not its not a quadratic equation because it is not equating to any thing.

For a quadratic equation it must be eual to something

for e.g.

5x² + 5x =8

 Is it a quadratic equation
Yes it is a quadratic equation

 How to solve Quadratic Equation

Before this we should know why we solve and what will we get after solving this equation
Solving a Quadratic equation means getting the values which will satisfy this equation
and those values which satisfy the equations are  known as roots.

For a Quadratic equation there can be only two roots possible for a single quadratic equation
 Since quad means two so only two roots are possible for a quadratic equation.

Consider this example 


  
                              let  y = x² + 5x + 6 = 0 

 we need to solve it or we have to find the roots of this equation.

roots are those values of x at which y = 0

                                                       
                                                        y =  x² + 5x + 6=0
                                      =>x² + 5x + 6=0
                                      =>x² + 3x+2x + 6=0
                                    => x(x+3) +2(x+3)  =0
                                  => (x+3)(x+2)=0
so roots are x= -3 and -2

if we place these roots in equation in place of x we ger

         first take x = -3  
       then              LHS          (-3) ² + 5(-3) + 6
                                             9     -15    +6
                                             -6+6        
                                            = 0
LHS = RHS