Math Snap

STEP 1

1. The equation $z(z+8) = -18$ is a quadratic equation.
2. Solving the equation involves expanding and rearranging terms to form a standard quadratic equation.
3. The quadratic formula or factoring will be used to find the solutions.

STEP 2

1. Expand and rearrange the equation.
2. Solve the quadratic equation.
3. Verify the solutions.

STEP 3

Expand the left-hand side of the equation:
$$z(z + 8) = z^2 + 8z$$

STEP 4

Rearrange the equation to form a standard quadratic equation by moving all terms to one side:
$$z^2 + 8z + 18 = 0$$

STEP 5

Attempt to factor the quadratic equation. We look for two numbers that multiply to $18$ and add to $8$. Since factoring is not straightforward, we use the quadratic formula:
$$z = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}$$ where $a = 1$, $b = 8$, and $c = 18$.

STEP 6

Calculate the discriminant $b^2 - 4ac$:
$$b^2 - 4ac = 8^2 - 4 \cdot 1 \cdot 18 = 64 - 72 = -8$$

STEP 7

Since the discriminant is negative, the solutions are complex. Calculate the solutions using the quadratic formula:
$$z = \frac{{-8 \pm \sqrt{{-8}}}}{2}$$ $$z = \frac{{-8 \pm 2i\sqrt{2}}}{2}$$ $$z = -4 \pm i\sqrt{2}$$

Verify the solutions by substituting back into the original equation. Since the solutions are complex, they are valid as the equation involves real coefficients.
The solution set is:
$$\{-4 + i\sqrt{2}, -4 - i\sqrt{2}\}$$ The solution set is:
$$\boxed{\{-4 + i\sqrt{2}, -4 - i\sqrt{2}\}}$$