Question

Is there any real value of 'a' for which the equation x² + 2x + (a² + 1) = 0 has real roots?

Answer 1

Let quadratic equation x² + 2x + (a² + 1) = 0 has real roots.

Here, a = 1, b = 2 and c = (a² + 1)

As we know that `D = b^2 - 4ac`

Putting the value of a = 1, b = 2 and c = (a² + 1), we get

\[D = \left( 2 \right)^2 - 4 \times 1 \times \left( a^2 + 1 \right)\]

\[ = 4 - 4\left( a^2 + 1 \right)\]

\[ = - 4 a^2\]

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.