Question

\[2 x^2 + x + 1 = 0\]

Answer 1

Given:   

\[2 x^2 + x + 1 = 0\]

Comparing the given equation with the general form of the quadratic equation 

\[a x^2 + bx + c = 0\], we get

\[a = 2, b = 1\] and \[c = 1\].

Substituting these values in 

\[\alpha = \frac{- b + \sqrt{b^2 - 4ac}}{2a}\] and \[\beta = \frac{- b - \sqrt{b^2 - 4ac}}{2a}\] ,we get:

\[\alpha = \frac{- 1 + \sqrt{1 - 4 \times 2 \times 1}}{2 \times 2}\] and \[\beta = \frac{- 1 - \sqrt{1 - 4 \times 2 \times 1}}{2 \times 2}\]

\[\Rightarrow \alpha = \frac{- 1 + \sqrt{- 7}}{4}\]    and     \[\beta = \frac{- 1 - \sqrt{- 7}}{4}\]

\[\Rightarrow \alpha = \frac{- 1 + i\sqrt{7}}{4}\]   and   \[\beta = \frac{- 1 - i\sqrt{7}}{4}\]

\[\Rightarrow \alpha = - \frac{1}{4} + \frac{\sqrt{7}}{4}i\]  and   \[\beta = - \frac{1}{4} - \frac{\sqrt{7}}{4}i\]

Hence, the roots of the equation are \[\frac{- 1 \pm i\sqrt{7}}{4}\] .