We can solve the equation \((z + 7)^2 = 40\) by following these steps:

### Step 1: Take the square root of both sides
\[
\sqrt{(z + 7)^2} = \pm \sqrt{40}
\]
This simplifies to:
\[
z + 7 = \pm \sqrt{40}
\]
Since \(\sqrt{40} = \sqrt{4 \times 10} = 2\sqrt{10}\), the equation becomes:
\[
z + 7 = \pm 2\sqrt{10}
\]

### Step 2: Solve for \(z\)
Now, subtract 7 from both sides:
\[
z = -7 \pm 2\sqrt{10}
\]
Thus, the two solutions for \(z\) are:
\[
z = -7 + 2\sqrt{10} \quad \text{or} \quad z = -7 - 2\sqrt{10}
\]

These are the exact solutions.

Would you like further details or clarification on any part of this process?

Here are five related questions:
1. How do you simplify radical expressions like \(\sqrt{40}\)?
2. What are the properties of square roots when solving quadratic equations?
3. How do we handle both positive and negative roots when solving equations like this?
4. How would you solve \((z + a)^2 = b\) for any general \(a\) and \(b\)?
5. What are the methods to check if the solutions are correct?

**Tip:** Always remember to consider both the positive and negative roots when taking square roots in an equation.

Discover how to solve the quadratic equation (z + 7)^2 = 40 in this step-by-step guide. This tutorial explains the square root method, simplifying radical expressions, and handling both positive and negative roots. Suitable for students in Grades 8-10.

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