Question

\[\sqrt{2} x^2 + x + \sqrt{2} = 0\]

Answer 1

Given:

\[\sqrt{2} x^2 + x + \sqrt{2} = 0\]

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

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

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

Substituting these values in

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

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

\[\Rightarrow \alpha = \frac{- 1 + \sqrt{- 7}}{2\sqrt{2}}\] and \[\beta = \frac{- 1 - \sqrt{- 7}}{2\sqrt{2}}\]

\[\Rightarrow \alpha = \frac{- 1 + i\sqrt{7}}{2\sqrt{2}}\] and  \[\beta = \frac{- 1 - i\sqrt{7}}{2\sqrt{2}}\]

Hence, the roots of the equation are \[\frac{- 1 \pm i\sqrt{7}}{2\sqrt{2}}\].