To solve the equation $5(x + 11)^2 = -50$, follow these steps:

1. Divide both sides by 5: $(x + 11)^2 = -10$

2. Notice that the right side is negative. Since a square of a real number cannot be negative, this indicates there are no real solutions.

3. If we allow complex solutions, we can continue: $(x + 11)^2 = -10$ Taking the square root of both sides gives: $x + 11 = \pm \sqrt{-10}$ This simplifies to: $x + 11 = \pm i\sqrt{10}$

4. Isolate $x$: $x = -11 \pm i\sqrt{10}$

Thus, the solutions are: $x = -11 + i\sqrt{10} \quad \text{and} \quad x = -11 - i\sqrt{10}$

Summary of Solutions

• The equation has no real solutions.
• The complex solutions are $x = -11 + i\sqrt{10}$ and $x = -11 - i\sqrt{10}$.

Would you like further details or explanations on any part of this solution? Here are some related questions:

1. What are imaginary numbers and how do they work?
2. Can you explain how to solve quadratic equations?
3. What is the significance of complex solutions in mathematics?
4. How do you graph complex numbers?
5. What are the applications of complex numbers in real-world scenarios?

Tip: Always check if a quadratic equation could yield complex solutions by looking for negative values under the square root.