Question

if alpha nd beta are the roots of quadratic equation ax2+bx+c=0 then the quadratic equation of ax2 +bx(x+1)+c(x+1)2=0​

Answer 1

Answer:

Step-by-step explanation:

Since α and β are the roots of ax  

2

+bx+c=0,

therefore

α+β=−  

a

b

​  

,αβ=  

a

c

​  

 

The equation ax  

2

−bx(x−1)+c(x−1)  

2

=0 can be written as x  

2

(a−b+c)+(b−2c)x+c=0

Sum of the roots of this equation is  

S=−  

a−b+c

b−2c

​  

=  

a−b+x

−b+2c

​  

=  

1−  

a

b

​  

+  

a

c

​  

 

−  

a

b

​  

+  

a

2c

​  

 

​  

 

⇒S=  

1+α+β+αβ

α+β+2αβ

​  

=  

α+1

α

​  

+  

β+1

β

​  

 

Product of the roots =  

a−b+c

c

​  

 

⇒P=  

1−  

a

b

​  

+  

a

c

​  

 

a

c

​  

 

​  

 

⇒P=  

1+α+β+αβ

αβ

​  

=  

α+1

α

​  

.  

β+1

β

​  

 

Thus, ax  

2

−bx(x−1)+c(x−1)  

2

=0

has  

α+1

α

​  

,  

β+1

β

​  

 as its two roots.