Question:
If $a$ and $b$ are roots of the equation $x^{2}-x+1=0$, then write the value of $a^{2}+b^{2}$.
Solution:
Given: $x^{2}-x+1=0$
Also, $a$ and $b$ are the roots of the equation.
Then, sum of the roots $=a+b=-\left(\frac{-1}{1}\right)=1$
Product of the roots $=a b=\frac{1}{1}=1$
$\therefore(a+b)^{2}=a^{2}+b^{2}+2 a b$
$\Rightarrow 1^{2}=a^{2}+b^{2}+2 \times 1$
$\Rightarrow a^{2}+b^{2}=1-2=-1$
$\Rightarrow a^{2}+b^{2}=-1$
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