Quadratic equation

Content-

  Quadratic equation:

format of the equation 
                   ax2 + bx + c = 0 ——— (1)
condition:
a , b , c R & a≠0 is known as quadratic equation.
here a , b , c are known as coefficient of equation (1)

Method to find roots of equation:

We know. 
ax2 + bx + c = 0
this equation can be written as—
x2 + (b/a)x + c/a = 0  , a≠0
x2 + (b/a)x = -c/a 
adding  both side 
( b/2a)
then 
x2 + (b/a)x + ( b/2a)2 = 
-c/a + ( b/2a)2
You observe in left side it is perfect square
 
(x+ b/2a)2 = ( b/2a)2 – c/a
(x+ b/2a)2  =  (b2 – 4ac)/4a2
Let assume
                      D = b2 – 4ac
x+ b/2a  =  ± √D/2a
          x = -b/2a ± √D/2a
x = (-b ± √D)/2a
case-1  using ‘+’ sign
        X1 = (-b + √D)/2a
Case-2  using ‘-’ sign
          X2 = (-b – √D)/2a
Here  X1 &  X2  are roots of equation (1)
Nature of roots : 
it is depends on  D.
  
  • if D < 0   complex roots or non real roots.
⦿ if D > 0  real roots 
                     X1 = (-b + √D)/2a
                   &    X2 = (-b – √D)/2a
Then   ax2 + bx + c = a(x-x1)(x-x2)
⦿ if D = 0   equal roots
                    X1 =  X2 = -b /2a
                  
  • some important points :
If a , b , c Q
                 D  = perfect square of a rational number .  then both roots are rational.
                  D  ≠ perfect square , then both roots are irrational.
⦿ if a , b , c Q  & p+iq  is one of root (q ≠ 0) then other root must be p-iq. So both roots are conjugate to each other. ( i = √-1 )
  • if a , b , c Q & p+√q  is one of root then other root must be p-√q. So both roots are conjugate to each other.
      • if a=1 & b , c I  and roots are rational number , then both roots must an integer.
  • if  quadratic equation  has more than two roots , then equation  becomes an identity.
   i.e  a=b=c=0.
Relation between roots & coefficients
If a & β are the roots of equation   ax2 + bx + c = 0  , then
1. a+ β = -b/a ( sum of roots)
2.  a .β = c/a ( product of roots)
3. An equation has roots  a  &  β  is
                     (x-a).(x- β) = 0
i.e     x2 – (a + β )x + a .β = 0
——————————————————
 4 . a2+ β 2 = (a + β )2_ 2a
 5 . a – β = √[ (a + β )2_ 4 a
 6 . a2 β 2 = (a + β ) (a – β )
 7 . a3+β 3=(a+β)3_3a(a+β)
 8 . a4+β 4=(a2+β 2 )2_ 2a2 . β 2