KCET · Maths · Quadratic Equation
If \( \alpha \) and \( \beta \) are roots of the equation \( \chi^{2}+x+1=0 \) then \( \alpha^{2}+\beta^{2} \) is
- A \( \frac{-1+i \sqrt{3}}{2} \)
- B \( -1 \)
- C \( 11 \)
- D \( \frac{-1-i \sqrt{3}}{2} \)
Answer & Solution
Correct Answer
(B) \( -1 \)
Step-by-step Solution
Detailed explanation
(B)
\(x^{2}+x+1=0\)
\(\Rightarrow x=\omega, \beta=\omega^{2}\) where \(\omega\) is the cube root of unity
\(\alpha^{2}+\beta^{2}=\omega^{2}+\omega^{4}\)
\(=\omega^{2}+\omega\)
\(=-1\)
\(x^{2}+x+1=0\)
\(\Rightarrow x=\omega, \beta=\omega^{2}\) where \(\omega\) is the cube root of unity
\(\alpha^{2}+\beta^{2}=\omega^{2}+\omega^{4}\)
\(=\omega^{2}+\omega\)
\(=-1\)
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