Quadratic Equation:All Quicker tricks for super fast answer

General form of Quadratic Equation 

 aX²+bX+c=0

Quicker Math tricks 

Quadratic Equation has two roots.

Usually we denote the two roots of Quadratic Equation by the symbol α and  β

In the equation 

aX²+bX+c=0 if  α and  β are the two roots then what is the value of α+ β?

Quicker tricks 
 α+ β=-b/a

Example-1:if α and β are the two roots of 3X²+2X+1=0 then what is the value of α+ β?

Then α+ β =-b/a

                  =-2/3

Example-2:if α and β are the two roots of 3X²+2X+1=0 then what is the value of αβ?

3X²+2X+1=0 if α and β are the two roots

           αβ=c/a

                =1/3

Example-3:if α and β are the two roots of 3X²+2X+1=0 then what is the value of 1/α+1/β?

3X²+2X+1=0 if α and β are the two roots

1/α+1/β =-b/c

              =-2/1=-2

Example-4:if α and β are the two roots of 3X²+2X+1=0 then what is the value of α²+β²?

3X²+2X+1=0 if α and β are the two roots

α²+β² =(α+ β)² -2αβ

          =(b² -2ac)/a²

          =(2² -2×3×1)/3²

          =(4-6)/9=-2/9

If α and β are the two roots of 
aX²+bX+c=0 then 
α-²+β-²=(b² - 2ac)/c²

Example-5:if α and β are the two roots of 3X²+2X+1=0 then what is the value of α-²+β-²?

α-²+β-²=(b² - 2ac)/c²

            

             =(2² -2×3×1)/1²

             =(4-6)/1=-2/1

             =-2

Example-6:if α and β are the two roots of 3X²+2X+1=0 then what is the value of 

(α+1)(β+1)?

(α+1)(β+1)=αβ+α+ β+1
=(a+c-b)/a

=(3+1-2)/3

=2/3

If α and β are the two roots of 
aX²+bX+c=0 and the two roots are multiplicative inverse to each other then

αβ=1

c/a=1

c=a

If α and β are the two roots of 

7X²+5X+c=0 and the two roots are multiplicative inverse to each other then what is the value of c?

c=7

If α and β are the two roots of 
aX²+bX+c=0 and the two roots are additive inverse  to each other then

α+β=0

-b/a=0

b=0

Example :

If α and β are the two roots of 

5X²+(k+3)X+1=0 and the two roots are additive inverse to each other then what is the value of k?

Solution -

K+3=0

k=-3

If the product and sum of two roots of aX²+bX+c=0 are same

Then -b=c

If the product and sum of two roots of 9X²+pX+4=0 are same then what is the value of p?

Solution -

P=-4

If one root of aX²+bX+c=0 is n times the other root then

nb²=(n+1)² ac

Example:If one root of pX²+2X+1=0 is 3 times the other root then what is the value of p?

Solution -

3(2)²=(3+1)² p×1

12=16p

p=12/16=3/4

Statistics:How to find mean?

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