Question

\[\sqrt{5} x^2 + x + \sqrt{5} = 0\]

Answer 1

Given: 

\[\sqrt{5} x^2 + x + \sqrt{5} = 0\]

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

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

\[a = \sqrt{5} , b = 1\] and \[c = \sqrt{5}\].

Substituting these values in

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

\[\alpha = \frac{- 1 + \sqrt{1 - 4 \times \sqrt{5} \times \sqrt{5}}}{2\sqrt{5}}\]   and \[\beta = \frac{- 1 - \sqrt{1 - 4 \times \sqrt{5} \times \sqrt{5}}}{2\sqrt{5}}\]

\[\alpha = \frac{- 1 + \sqrt{- 19}}{2\sqrt{5}}\] and  \[\beta = \frac{- 1 - \sqrt{- 19}}{2\sqrt{5}}\]

\[\alpha = \frac{- 1 + i\sqrt{19}}{2\sqrt{5}}\] and \[\beta = \frac{- 1 - i\sqrt{19}}{2\sqrt{5}}\]

Hence, the roots of the equation are 

\[\frac{- 1 \pm i\sqrt{19}}{2\sqrt{5}}\].