If a, b are the roots of the equation
Question:

If $a, b$ are the roots of the equation $x^{2}+x+1=0$, then $a^{2}+b^{2}=$

(a) 1

(b) 2

(c) −1

(d) 3

Solution:

(c) −1

Given equation: $x^{2}+x+1=0$

Also, $a$ and $b$ are the roots of the given equation.

Sum of the roots $=a+b=\frac{-C \text { oefficient of } x}{C \text { oefficient of } x^{2}}=-\frac{1}{1}=-1$

Product of the roots $=a b=\frac{C \text { onstant term }}{C \text { oefficient of } x^{2}}=\frac{1}{1}=1$

$\therefore(a+b)^{2}=a^{2}+b^{2}+2 a b$

$\Rightarrow(-1)^{2}=a^{2}+b^{2}+2 \times 1$

$\Rightarrow 1-2=a^{2}+b^{2}$

$\Rightarrow a^{2}+b^{2}=-1$