Question: Consider the following equation.
$4x^2 - 9 = -409$
Find all real and complex solutions to the equation. If multiple solutions exist, enter the
solutions using a comma-separated list. If no solution exists, enter DNE.
$x = \{ \}$
1. **State the problem:** Solve the equation $$4x^2 - 9 = -409$$ for all real and complex values of $x$.
2. **Rewrite the equation:** Add $9$ to both sides to isolate the quadratic term:
$$4x^2 - 9 + 9 = -409 + 9$$
which simplifies to
$$4x^2 = -400$$
3. **Divide both sides by 4:**
$$\cancel{4}x^2 = \frac{-400}{\cancel{4}}$$
which simplifies to
$$x^2 = -100$$
4. **Solve for $x$ by taking the square root of both sides:**
$$x = \pm \sqrt{-100}$$
5. **Simplify the square root of a negative number:**
$$\sqrt{-100} = \sqrt{100} \times \sqrt{-1} = 10i$$
where $i$ is the imaginary unit.
6. **Write the final solutions:**
$$x = \pm 10i$$
**Answer:**
$$x = 10i, -10i$$