Consider the equation,

\(\displaystyle\frac{{1}}{{{2}{z}^{{2}}}}=−{3}\)

Left hand side \(\displaystyle=\frac{{1}}{{{2}{z}^{{2}}}}\) and Right hand side = - 3.

From above equation,

Left hand side \(\displaystyle=\frac{{1}}{{{2}{z}^{{2}}}}\) is always positive for any values of z.

Fight hand side = -3 is negative .

To get,

Left hand side is not equal to the right hand side.

Since left hand side is always positive for any values of z except if z is zero or undefined , it can be never equal a negative number and right hand side is negative. Then,

The equation \(\displaystyle\frac{{1}}{{{2}{z}^{{2}}}}=−{3}\) has no solution.

\(\displaystyle\frac{{1}}{{{2}{z}^{{2}}}}=−{3}\)

Left hand side \(\displaystyle=\frac{{1}}{{{2}{z}^{{2}}}}\) and Right hand side = - 3.

From above equation,

Left hand side \(\displaystyle=\frac{{1}}{{{2}{z}^{{2}}}}\) is always positive for any values of z.

Fight hand side = -3 is negative .

To get,

Left hand side is not equal to the right hand side.

Since left hand side is always positive for any values of z except if z is zero or undefined , it can be never equal a negative number and right hand side is negative. Then,

The equation \(\displaystyle\frac{{1}}{{{2}{z}^{{2}}}}=−{3}\) has no solution.