Question

\[x^2 + \frac{x}{\sqrt{2}} + 1 = 0\]

Answer 1

Given equation: 

\[x^2 + \frac{x}{\sqrt{2}} + 1 = 0\]

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

\[a x^2 + bx + c = 0\] ,we get 

\[a = 1, b = \frac{1}{\sqrt{2}}\] 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:

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

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

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

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

Hence, the roots of the equation are 

\[\frac{- 1 \pm i\sqrt{7}}{2\sqrt{2}}\] .