Answer
The two solutions of the equation ${{x}^{2}}+2x+5=0$ are imaginary.
Work Step by Step
$a{{x}^{2}}+bx+c=0$, the radicand ${{b}^{2}}-4ac$ is known as the discriminant.
1. If the value of the discriminant is zero then there is one repeated solution that is rational.
2. When the value of the discriminant is positive, there are two different real rational solutions if the discriminant is a perfect square whereas there are two different real irrational solutions if the discriminant is not a perfect square.
3. If the value of the discriminant is negative then the solutions are imaginary.
Consider the expression, ${{x}^{2}}+2x+5=0$.
Here,
$\begin{align}
& a=1 \\
& b=2 \\
& c=5 \\
\end{align}$
So, the value of the discriminant is:
$\begin{align}
& {{b}^{2}}-4ac={{2}^{2}}-4\left( 1 \right)\left( 5 \right) \\
& =4-20 \\
& =-16
\end{align}$
The term $-16$ is negative. So, the solutions will be imaginary.
Thus, the two solutions of the equation ${{x}^{2}}+2x+5=0$ are imaginary.
