https://file.aiquickdraw.com/mathbot/file/1726766594600kzerlfxd.jpg

The equation in the image is:

\[
\sqrt{x + 3} = x - 1
\]

Let's solve it step by step.

### Step 1: Square both sides to eliminate the square root.

\[
(\sqrt{x + 3})^2 = (x - 1)^2
\]

This simplifies to:

\[
x + 3 = (x - 1)^2
\]

### Step 2: Expand the right-hand side.

\[
(x - 1)^2 = x^2 - 2x + 1
\]

So the equation becomes:

\[
x + 3 = x^2 - 2x + 1
\]

### Step 3: Move all terms to one side.

\[
0 = x^2 - 2x - x + 1 - 3
\]

This simplifies to:

\[
0 = x^2 - 3x - 2
\]

### Step 4: Factor the quadratic equation.

\[
x^2 - 3x - 2 = (x - 2)(x + 1)
\]

### Step 5: Solve for \(x\).

Set each factor equal to zero:

\[
x - 2 = 0 \quad \text{or} \quad x + 1 = 0
\]

Thus, the solutions are:

\[
x = 2 \quad \text{or} \quad x = -1
\]

### Step 6: Check for extraneous solutions.

Since we squared both sides, we need to check which of these solutions satisfy the original equation.

- For \(x = 2\):

\[
\sqrt{2 + 3} = 2 - 1 \quad \Rightarrow \quad \sqrt{5} \neq 1
\]

So, \(x = 2\) is not a solution.

- For \(x = -1\):

\[
\sqrt{-1 + 3} = -1 - 1 \quad \Rightarrow \quad \sqrt{2} \neq -2
\]

Thus, \(x = -1\) is also not a solution.

### Conclusion:
There are **no real solutions** to this equation.

---

Would you like a deeper explanation or have any questions? Here are some related questions to explore further:

1. What are extraneous solutions, and why do they occur when solving equations?
2. How can you graphically interpret the equation \( \sqrt{x + 3} = x - 1 \)?
3. Can we modify the equation to find complex solutions?
4. What other methods can we use to solve radical equations?
5. How does squaring both sides of an equation affect the solution set?

**Tip:** Always check for extraneous solutions when solving radical equations by substituting them back into the original equation.

Solve the equation \(\sqrt{x + 3} = x - 1\).

This page provides a detailed solution to the radical equation \(\sqrt{x + 3} = x - 1\). Learn how to solve it step-by-step by squaring both sides and factoring the quadratic equation. Suitable for students in Grades 9-11.

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