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Question
The equation of the smallest degree with real coefficients having 1 + i as one of the roots is
Options
\[x^2 + x + 1 = 0\]
\[x^2 - 2x + 2 = 0\]
\[x^2 + 2x + 2 = 0\]
\[x^2 + 2x - 2 = 0\]
Solution
\[x^2 - 2x + 2 = 0\]
We know that, imaginary roots of a quadratic equation occur in conjugate pair.
It is given that, 1 + i is one of the roots.
So, the other root will be \[1 - i\] .
Thus, the quadratic equation having roots 1 + i and 1 - i is,
\[x^2 - \left( 1 + i + 1 - i \right)x + \left( 1 + i \right)\left( 1 - i \right) = 0\]
\[ \Rightarrow x^2 - 2x + 2 = 0\]
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