Question

Find the discriminant.

z^2 + z + 6 = 0
How many real solutions does the equation have?

Answer 1

To find the discriminant of the quadratic equation \( z^2 + z + 6 = 0 \), we can use the formula for the discriminant, which is given by:

\[
D = b^2 - 4ac
\]

In the equation \( z^2 + z + 6 = 0 \), we identify the coefficients as follows:
- \( a = 1 \)
- \( b = 1 \)
- \( c = 6 \)

Plugging these values into the discriminant formula:

\[
D = (1)^2 - 4(1)(6)
\]
\[
D = 1 - 24
\]
\[
D = -23
\]

The discriminant \( D = -23 \) is less than zero.

Since the discriminant is negative, this means the quadratic equation has no real solutions. Instead, it has two complex solutions.