METHODE OF FACTORIZATION FOR QUADRATIC EQUATON WITHOUT GUESSING

QUADRATIC EQUATION

The standard form of the quadratic equation is

                     

                                         Ax²+Bx +C=0     (A≠0)

Where A , B, C are the numerical coefficients and "x" is the unknown variable.

ZERO FACTOR PRINCIPLE

IF AB = 0 then either A=0 or B=0

This property is called zero factor property or zero factor principle.

Note* this property work only when the product is equal to zero.

Consider AB =6, Then we can't say that A=6 or B=6, here we could have A=2 and B=3 for instance.

METHODE FOR MAKING FACTORS

        To solve a quadratic equation by factoring we first must move all the terms over to one side of the equation. Doing this serves two purposes. First, it puts the quadratics into a form that can be factored. Secondly, and probably more importantly, in order to use the zero factor property, we MUST have a zero on one side of the equation. If we don’t have a zero on one side of the equation we won’t be able to use the zero factor property. This method is used when B is even.

For solving quadratic equations of the form 

                                  Ax²+Bx+C=0_____________(1)

Then, we have to find some M and N Such that 

B = M+N___________(2)

A*C = M* N________(3)

Here by using the following method we can find M and N very easily

Now consider

 M = B/ 2+ U________(4).  and

 N = B/2 - U_________(5)

Using these values in (3) we got 

⇒( B/2+U)(B/2 -U) = A*C =K ( say)

⇒B²/4 -U² = K

⇒U²= B²/4 - K

⇒U= √(B²/4- K)

Using the value of U in (4) and (5) we got the values of M and N , then by using values of M and N in (1) we can easily factorize the equation.

Exp: x²+8x+12=0

sol:

x²+8x+12=0____________(1)

Here M+N = B  And M * N = A*C

I.e. M+ N = 8 and M*N = 12         

Now to find The values of M and N 

Consider 

M = 8/2 + U = 4 + U____________(2)

N = 8/2 - U = 4 - U_____________(3)

M*N = 12

⇒(4-U)(4+U)= 12

⇒4²-U²=12

⇒16-U²=12

⇒U²=16-12=4

⇒U =√4 =2

Using value of U in (2) and (3) we got

M= 6 and N = 2

Then using these values in (1) we 

⇒x²+(6+2)x+12=0

⇒x²+6x+2x+12=0

⇒x(x+6)+2(x+6)=0

⇒(x+6)(x+2)=0

⇒x+6=0 OR x+2 =0 

⇒x=-6 OR x= -2

Exp: 8x²+16x+6=0

sol:

Let M+N = 16

And M*N = 8*6=48

Now consider M = 16/2 + U and N = 16/2 - U

⇒(8+U)(8-U)=48

⇒8² - U²= 48

⇒64  - 48 = U²

⇒U= √16= 4

So, M = 8+4=12

And N = 8-4 = 4

So our equation becomes 

⇒x²+(12+4)x+48=0

⇒x²+ 12x+4x + 48= 0

⇒x(x+12)+4(x+12)=0

⇒(x+12)(x+4)=0

⇒x+12=0 or X+ 4 = 0 

⇒x = -12 or x= - 4

EXERCISE:

1. x²+6x+8=0

2. 4x²+12x+9=0

For any queries comment below...

Thanks

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$$\text{HEREIf}ab=0\text{then either}a=0\text{and/or}b=0$$

Is zero an even or an odd number

Is zero an even number or an odd number or both.
Let's see both cases.

When a number is even?
A number is said to be even if
it's divisible by 2 or it's a multiple of 2.

0÷2=0
0×2=0

So 0 is divisible by 2 as well as a multiple of 2.
Hence 0 is an even Number.

Now Let's find out why '0' is not an odd number:
Any integer n is considered as an odd number if 'n+1' is an even number.
But '0' does not fulfill this condition.
0+1=1 and 1 is not an even number.
Hence '0' is not an odd number.


Let's see some arguments which proves that '0' is an even number.

• Between every consecutive odd integers there is an even integer.  '0' lies between -1 & 1 which are odd integers, so '0' is an even number.
• An Odd number is always followed by an Even number.
  But '0' is followed by 1 which is an odd number.
  Hence '0' can't be an Odd number.

If  you know any reasons which proves '0' is an odd number.
COMMENT BELOW


                                                      
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Tricky Methods to find the square of a number quickly ( Vedic math )

What's square of a number?
When we multiply a number with itself the resultant number is called square of that number.

Finding square by adding consecutive odd number:-

1² = 1

2²=1+3=4

3²=1+3+5=9

4²=1+3+5+7=16

5²=1+3+5+7+9=25

....... So on

Finding Square by using a Base:-

Base:- Bases are generally the powers of 10.

i.e 10,100,1000,10000,........ etc

Case_I:(if the number is less then The Base)

Exp:(1)

let the number is 96

   Here the base is 100 

   Here the number is less then from base by 04

  Then ,

             96²= 96-04 | (04)²

                        = 9216
So the answers is 9216

Exp:(2)

Let the number is 992

 Here the base is 1000

 Here the number is less then from base by 008

Then ,

          992²= 992-008|(008)²

                          = 984064
So the answers is 984064

Exp(3):

Let the number is 9975

  Here the base is 10000

  Here the number is less then base by 0025

Then,

         9975²=9975-0025/(0025)²

                             =99500625
So the answers is 99500625

Case_II:(If the number is greater then the Base)

Exp:(1)

Let the number is 17

  Here the base is 10

  And the number is greater than the base by 07

Then,

         17² = 17+7|(7)²

                 = 24|49

                 =289 
( here the base is the 1st power of 10 so 4 is carried towards the left and added with 24)

So the answers is 289

Exp:(2)

Let the number is 115

  Here the base is 100

  And the number is greater then from base by 15

Then, 

          115²= 115+15|(015)²

                          =130225
So the answers is 130225

Exp(3):

Let The number is 1025

   Here the base is 1000

   And the number is greater than base by 25

Then,

         1025²= 1025+25|(025)²

                          = 1050625
So the answers is 1050625

If the unit digit of the number is "5"

Let MN be the number where N is 5

Then ,

            MN² =M*(M+1)|5²

Exp(1):

Let the number is 45

 45²= 4*5|(5)²

          =2025
So the answers is 2025

Exp(2):

Let the number be 95

   95²= 9*10| 5²

            =9025
So the answers is 9025

Exp(3):

Let the number be 125

 125²= 12*13|5²

           =15625
So the answers is 15625



Dwanda yoga / Duplex combination

The Dwandwa Yoga or 'Duplex Combination' can be used for general purpose squaring.
To proceed further we need to know the Dwandwa of certain numbers.

D( a ) = a²
D( ab ) = 2ab
D( abc ) = 2ac + b²
D( abcd ) = 2ad + 2bc
D( abcde ) = 2ae + 2bd + c²
D( abcdef ) = 2af + 2be + 2cd      and so on....

As we can see above, D of any number is the sum of square of the middle number and two times the product of the other pairs.
Square of a number is given by

( ab )² = D( a ) | D( ab ) | D( b )
( abc )² = D( a ) | D( ab ) | D(abc) | D( bc ) | D( c )
( abcd )² = D(a) | D(ab) | D(abc) | D(abcd) | D(bcd) | D(bc) | D (c)

Exp(1):
Let the number is 23
Then,

      (23)² = (ab)² = D(a) | D(ab) | D(b)
                               = 4     |    12    |    9
Since Dwanada must have only one digit, we carry over '1' of '12' to LHS.
Therefore it becomes     4 |₁ 2| 9
So the  answer is 529

Exp(2):
Let the number is 527
Then,

    (527)² = ( abc )²
           = D( a ) | D( ab ) | D(abc) | D( bc ) | D( c )
           =   25   |     20    |     74   |    28    |   49
           =   25   |₂     0    |₇     4   |₂    8    |₄   9
          =   277729
So the answer is 277729

For any query Comment below:-
Thank you.....
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