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Question
If a, b are the roots of the equation \[x^2 + x + 1 = 0, \text { then } a^2 + b^2 =\]
Options
1
2
-1
3
Solution
−1
Given equation:
\[x^2 + x + 1 = 0\]
Also,
\[a\] and \[b\] are the roots of the given equation.
Sum of the roots = \[a + b = \frac{- \text { Coefficient of }x}{\text { Coefficient of } x^2} = - \frac{1}{1} = - 1\]
Product of the roots = \[ab = \frac{\text { Constant term }}{\text { Coefficient of } x^2} = \frac{1}{1} = 1\]
\[\therefore (a + b )^2 = a^2 + b^2 + 2ab\]
\[ \Rightarrow ( - 1 )^2 = a^2 + b^2 + 2 \times 1\]
\[ \Rightarrow 1 - 2 = a^2 + b^2 \]
\[ \Rightarrow a^2 + b^2 = - 1\]
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