Question

solve by using the quadratic formula.
$z^2 = -4z - 6$
separate your answers with commas, if necessary.
express the solution set in exact simplest form.
the solution set is { }.

Explanation:

Step1: Rewrite in standard form

First, we rewrite the equation \( z^2 = -4z - 6 \) in standard quadratic form \( az^2 + bz + c = 0 \).
Adding \( 4z \) and \( 6 \) to both sides, we get:
\( z^2 + 4z + 6 = 0 \)
Here, \( a = 1 \), \( b = 4 \), and \( c = 6 \).

Step2: Apply the quadratic formula

The quadratic formula is \( z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
Substituting the values of \( a \), \( b \), and \( c \):
First, calculate the discriminant \( D = b^2 - 4ac \):
\( D = (4)^2 - 4(1)(6) = 16 - 24 = -8 \)
Then, substitute into the quadratic formula:
\( z = \frac{-4 \pm \sqrt{-8}}{2(1)} \)
Simplify \( \sqrt{-8} \):
\( \sqrt{-8} = \sqrt{8} \cdot \sqrt{-1} = 2\sqrt{2}i \) (since \( \sqrt{-1} = i \))
So,
\( z = \frac{-4 \pm 2\sqrt{2}i}{2} \)
Simplify the fraction by dividing numerator and denominator by 2:
\( z = -2 \pm \sqrt{2}i \)

Answer:

\( -2 + \sqrt{2}i, -2 - \sqrt{2}i \)