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

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

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

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

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