https://file.aiquickdraw.com/mathbot/file/1726323388186nb28o66r.jpg

The equation from the image is:

\[
\sqrt{3 + x} - \sqrt{x + 7} = -2
\]

To solve this, follow these steps:

### Step 1: Isolate one of the square roots.
We will isolate the square root on the left:

\[
\sqrt{3 + x} = \sqrt{x + 7} - 2
\]

### Step 2: Square both sides.
Squaring both sides to remove the square roots:

\[
(\sqrt{3 + x})^2 = (\sqrt{x + 7} - 2)^2
\]

Simplifying both sides:

\[
3 + x = (x + 7) - 4\sqrt{x + 7} + 4
\]

This simplifies further to:

\[
3 + x = x + 11 - 4\sqrt{x + 7}
\]

### Step 3: Isolate the square root again.
Subtract \(x + 11\) from both sides:

\[
3 + x - (x + 11) = -4\sqrt{x + 7}
\]

This simplifies to:

\[
-8 = -4\sqrt{x + 7}
\]

### Step 4: Solve for \(x\).
Divide both sides by -4:

\[
2 = \sqrt{x + 7}
\]

Square both sides again:

\[
4 = x + 7
\]

Finally, subtract 7 from both sides:

\[
x = -3
\]

### Step 5: Check the solution.
Substitute \(x = -3\) back into the original equation:

\[
\sqrt{3 + (-3)} - \sqrt{-3 + 7} = -2
\]

Simplifying both sides:

\[
\sqrt{0} - \sqrt{4} = -2
\]

\[
0 - 2 = -2
\]

This is true, so the solution is \(x = -3\).

Would you like more details or have any further questions?

Here are five related questions for further exploration:
1. How can you solve equations involving cube roots?
2. What are some methods to solve radical equations with multiple terms?
3. How can extraneous solutions arise when squaring both sides of an equation?
4. What are the properties of square root functions?
5. How would the solution change if the constants were different in the equation?

**Tip:** Always check for extraneous solutions when solving radical equations, as squaring can introduce false solutions.

Solve the equation: sqrt(3 + x) - sqrt(x + 7) = -2

Learn how to solve the radical equation sqrt(3 + x) - sqrt(x + 7) = -2. This step-by-step solution includes isolating square roots, squaring both sides, and checking for extraneous solutions. Suitable for high school students.

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