if alpha and beta are the zeros of the quadratic polynomial f(x)=x²-px+q then find the value of alpha²beta²+alpha beta ²
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if alpha and beta are the zeros of the quadratic polynomial f(x)=x²-px+q then find the value of alpha²beta²+alpha beta ²
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Solution
We know (a+b)² = a² + b² + 2ab
So a² + b² = (a+b)² - 2ab
Similarly α² + β² = (α+β)² - 2αβ
Put value of (α+β) and αβ we will get
α² + β² = (α+β)² - 2αβ
α² + β² = p² - 2q
Answer : α² + β² = p² - 2q
Consider the given polynomial
f(x)=x
2
−px+q
α and βare the zeros of given polynomial
We know that,
Sumofzeros=−
cofficentofx
2
cofficentofx
α+β=
1
−(−p)
α+β=p
Productofzeros=
cofficentofx
2
constant
α×β=
1
q
α×β=q
Then,
Part (1):-
α
2
+β
2
=(α+β)
2
−2αβ
α
2
+β
2
=p
2
−2q
Part (2):-
α
1
+
β
1
=
αβ
α+β
=
q
p
Hence, this is the answer.
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