To solve the equation $(x + 3)^2 = -25$ using the square root property, follow these steps:

Step 1: Isolate the square term

The equation is already in the form $(x + 3)^2 = -25$. This matches the structure required for using the square root property.

Step 2: Take the square root of both sides

To solve for $x + 3$, take the square root of both sides of the equation:

$\sqrt{(x + 3)^2} = \sqrt{-25}$

On the left side, $\sqrt{(x + 3)^2} = x + 3$.

However, on the right side, $\sqrt{-25}$ involves an imaginary number since the square root of a negative number is imaginary. We rewrite this as:

$\sqrt{-25} = 5i$

Thus, the equation becomes:

$x + 3 = \pm 5i$

Step 3: Solve for $x$

Now solve for $x$ by subtracting 3 from both sides:

$x = -3 \pm 5i$

Final Answer:

The solutions are:

$x = -3 + 5i \quad \text{or} \quad x = -3 - 5i$

These are the two complex solutions to the equation.

---

Would you like further details or clarifications?

Here are 5 related questions for deeper understanding:

1. What are the steps to solve a quadratic equation using the quadratic formula?
2. How does the square root property help solve quadratic equations?
3. What is an imaginary number, and why is $i = \sqrt{-1}$?
4. How can you graphically represent complex solutions of quadratic equations?
5. How do real and complex solutions differ in quadratic equations?

Tip:

When dealing with square roots of negative numbers, remember that they result in imaginary numbers, and the basic imaginary unit is $i = \sqrt{-1}$.