Question

if two roots of the quadratic equation x²+px+q=0 are alpha and beta,then let us express the value of alpha³+ beta³ and 1/ alpha+1/ beta in terms of p and q​

Answer 1

Answer:

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Step-by-step explanation:

If α,β are the roots of the equation x  

2

−px+q=0,

⇒ sum of the roots =α+β=p    ...(1)

Similarly, product of the roots=αβ=q    ....(2)

If  

β

α

​  

,  

α

β

​  

 are roots of the a quadratic equation, then the quadratic equation is given by,

(x−  

β

α

​  

)(x−  

α

β

​  

)=0

⇒x  

2

−(  

β

α

​  

+  

α

β

​  

)x+1=0

⇒αβx  

2

−(α  

2

+β  

2

)x+1=0

⇒αβx  

2

−((α+β)  

2

−2αβ)x+1=0    ....(3)

Substituting (1),(2)  in (3), we get

qx  

2

−(p  

2

−2q)x+q=0

⇒qx  

2

+(2q−p  

2

)x+q=0