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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