. If α and β are zeroes of x2 – 4x + 1, then 1/α + 1/β – αβ is
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Answer:
x
2
−4x+1
Comparing to the general quadratic equation,
ax^2+bx+cax
2
+bx+c
Where,
a= 1, b= -4, c= 1
Given that α and β are zeroes of the polynomial.
From the properties of the zeros,
\begin{lgathered}\alpha +\beta =-\dfrac{b}{a}=\dfrac{4}{1}=4\\\\ \alpha \cdot \beta=\dfrac{c}{a}=\dfrac{1}{1}=1\end{lgathered}
α+β=−
a
b
=
1
4
=4
α⋅β=
a
c
=
1
1
=1
Then,
=\dfrac{1}{\alpha}+\dfrac{1}{\beta}-\alpha\beta=
α
1
+
β
1
−αβ
=\dfrac{\alpha+\beta}{\alpha\beta}-\alpha\beta=
αβ
α+β
−αβ
=\dfrac{4}{1}-1=
1
4
−1
=4-1=4−1
=3=3
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