Quadratic Equations

  MCQ's collected from all Engineering Entrance Examinations (Click here)


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.


 

 

 

 


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