Question:
Is there any real value of ' $a$ ' for which the equation $x^{2}+2 x+\left(a^{2}+1\right)=0$ has real roots?
Solution:
Let quadratic equation $x^{2}+2 x+\left(a^{2}+1\right)=0$ has real roots.
Here, $a=1, b=2$ and,$c=\left(a^{2}+1\right)$
As we know that $D=b^{2}-4 a c$
Putting the value of $a=1, b=2$ and, $c=\left(a^{2}+1\right)$, we get
$D=(2)^{2}-4 \times 1 \times\left(a^{2}+1\right)$
$=4-4\left(a^{2}+1\right)$
$=-4 a^{2}$
The given equation will have equal roots, if $D>0$
i.e. $-4 a^{2}>0$
$\Rightarrow a^{2}<0$
which is not possible, as the square of any number is always positive.
Thus, No, there is no any real value of $a$ for which the given equation has real roots.