https://file.aiquickdraw.com/mathbot/file/1729131037478j39za8qb.jpg

The equation provided is:

\[
\sqrt{v + 32} - \sqrt{v} = 2
\]

### Steps to solve:

1. **Isolate one of the square roots**:
   
   Add \(\sqrt{v}\) to both sides:

   \[
   \sqrt{v + 32} = \sqrt{v} + 2
   \]

2. **Square both sides** to eliminate the square roots:

   \[
   (\sqrt{v + 32})^2 = (\sqrt{v} + 2)^2
   \]

   Simplifying both sides:

   \[
   v + 32 = (\sqrt{v})^2 + 2 \cdot \sqrt{v} \cdot 2 + 2^2
   \]

   Which becomes:

   \[
   v + 32 = v + 4\sqrt{v} + 4
   \]

3. **Simplify the equation**:

   Subtract \(v\) from both sides:

   \[
   32 = 4\sqrt{v} + 4
   \]

   Subtract 4 from both sides:

   \[
   28 = 4\sqrt{v}
   \]

4. **Solve for \(\sqrt{v}\)**:

   Divide both sides by 4:

   \[
   \sqrt{v} = 7
   \]

5. **Square both sides** to solve for \(v\):

   \[
   v = 49
   \]

### Verify the solution:

Substitute \(v = 49\) back into the original equation:

\[
\sqrt{49 + 32} - \sqrt{49} = \sqrt{81} - \sqrt{49} = 9 - 7 = 2
\]

This satisfies the equation, so the solution is correct.

### Solution:

\[
v = 49
\]

Would you like more details on any step or have any questions?

### Follow-up questions:

1. How would the solution change if the equation was \(\sqrt{v + 32} + \sqrt{v} = 2\)?
2. What happens if the number inside the square root is negative?
3. How do we handle equations with two square roots on both sides in general?
4. What if there was a coefficient before one of the square roots?
5. How do you determine if an equation has no solution when dealing with square roots?

**Tip**: When dealing with square roots, always check for extraneous solutions caused by squaring both sides.

Solve for v, where v is a real number: sqrt(v + 32) - sqrt(v) = 2

Learn how to solve the equation sqrt(v + 32) - sqrt(v) = 2 using step-by-step methods. This involves isolating square roots, squaring both sides, and simplifying the equation to find the real solution. Suitable for students in Grades 10-12.

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