Question

if the equations X square + a x + B C is equal to zero and X square + bx + c is equal to zero have a common root then there are other routes that satisfy the equation is​

Answer 1

Answer:

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Step-by-step explanation:

For quadratic equation ax  

2

+bx+c=0  

Sum of roots =−  

a

b

​  

 

Product of roots =  

a

c

​  

 

Let x  

2

+ax+bc=0 have roots α and β:

α+β=−a......................(1)

αβ=bc...................(2)

Let x  

2

+bx+ac=0 have roots α and γ:

α+γ=−b..................(3)

αγ=ac..................(4)

Subtracting (3) from (1) ,we get

β−γ=b−a..................(5)

Dividing (2) and (3)

γ

β

​  

=  

a

b

​  

.................(6)

From (5) and (6)

β=b,γ=a

From (2)

α.b=bc⇒α=c

Now, equation having other roots is  

x  

2

−(β+γ)x+βγ=0

x  

2

−(a+b)x+ab..................(7)  

Since, β=b is the root of x  

2

+ax+bc=0

⇒b  

2

+ab+bc=0⇒a+b=−c

Putting this in (7)

we get x  

2

+cx+ab=0