### Important Formulas :

1. If    are real or complex numbers  then   is called a quadratic expression in    and   is called a quadratic equation. If      are all rational, then   is called a rational quadratic equation.

2.A real or complex number  is said to be a root  or solution of the quadratic equation  if 

3.The roots of the quadratic equation    are 

4.  is called the discriminate of the of the quadratic equation    .

5.If    are the roots of the quadratic equation    , then    and 

6.The quadratic equation whose roots are  and   is 

7.If     is irrational and if  is a root to the rational quadratic equation, then  is also a root.

8.If   is a root of the real quadratic equation, then  is also a root.

9.If   are real and , then the nature of the roots of equation     is as follows: If  , then the roots are real and distinct

10.If   are real and , then the nature of the roots of equation     is as follows: If , then the roots are real and equal. In this case each root is equal to  then   , and is called the double root or a repeated root

11.If   are real and , then the nature of the roots of equation     is as follows: If , then the roots are two non-real conjugate complex numbers.

12.If   are rational  and , then the nature of the roots of equation     is as follows: If , and is perfect square then the roots are rational and distinct.

13.If   are rational  and , then the nature of the roots of equation     is as follows: If , is not a perfect square then the roots are conjugate surds (irrationals.)

14.If   are rational  and , then the nature of the roots of equation     is as follows: If , then the roots are rational and equal.

15.If   are rational  and , then the nature of the roots of equation     is as follows: If ,  then the roots non-real conjugate complex numbers.

16.If   are rational  and , then the nature of the roots of equation     is as follows: If , and  is a perfect square, then the roots are integers.

17.Relation between the roots   of , where  , are real. .

18.Relation between the roots   of , where  , are real. 

19.Relation between the roots   of , where  , are real.  

20.Relation between the roots   of , where  , are real.  if 

21.Relation between the roots   of , where  , are real.  

22.Relation between the roots   of , where  , are real.  

23.Relation between the roots   of , where  , are real.  

24.Relation between the roots   of , where  , are real. 

25.Relation between the roots   of , where  , are real. 

26.Relation between the roots   of , where  , are real. 

27.Relation between the roots   of , where  , are real. 

28.Relation between the roots   of , where  , are real. 

30.Relation between the roots   of , where  , are real. 

31.Relation between the roots   of , where  , are real 

32.Relation between the roots   of , where  , are real 

33.For any quadratic equation , If   and   are of the same sign, then product of roots   is   the roots have the same sign.

34.For any quadratic equation , If   and    are of opposite signs , then product of roots   is   the roots will have opposite sign

35.For any quadratic equation , If both the roots are    then    will have the same sign.

36.For any quadratic equation , If the two roots are    then  will have the same sign different from the sign .

37.For any quadratic equation , If .then the roots are reciprocal to each other [product of the roots is 1]

38.For any quadratic equation , If . then the sum of the roots is equals to zero [the roots have the same absolute value i.e., the roots are in the form   ]

39.For any quadratic equation , If   then the roots are  and 

40.For any quadratic equation , If  i.e.,   then the roots are  and .

41.For any quadratic equation , If one root is zero then  .

42.For any quadratic equation , If   are integers and the roots are rational  numbers, then these roots must be integers.

43.For any quadratic equation , If   the roots are in the ratio , then .

44.For any quadratic equation , If one root is ,times the other ,  then .

45.For any quadratic equation , If one root is equal to the , power of the other root then 

46.For any quadratic equation , If one root is the square of the otherthen .

47.For any quadratic equation , If   and  have the same roots then 

48.For any quadratic equation , If the equations   and   have a common root then  and the common root is 

49.If the equation , has non real complex roots  , then   and  will have the same sign .

50.If the equation , has equal rootsthen  and  have the same sign .

51.If the equation , has real roots  and    then  and   are of opposite sign.

52.If the equation , has real roots  and    then   and     have the same sign.

for_QUODRATIC-EQUATIONS