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