https://file.aiquickdraw.com/mathbot/file/17254593119737atiqgno.jpg

The equation in the image is:

\[
\sqrt{3 - x} - \sqrt{7 + x} = \sqrt{16 + 2x}
\]

Let's solve this step by step.

### Step 1: Isolate one of the square root terms

Move \(\sqrt{7 + x}\) to the right-hand side:

\[
\sqrt{3 - x} = \sqrt{16 + 2x} + \sqrt{7 + x}
\]

### Step 2: Square both sides to eliminate the square root

Squaring both sides will help us remove the square root on the left side.

\[
(\sqrt{3 - x})^2 = (\sqrt{16 + 2x} + \sqrt{7 + x})^2
\]

This simplifies to:

\[
3 - x = (16 + 2x) + (7 + x) + 2\sqrt{(16 + 2x)(7 + x)}
\]

Simplify the equation:

\[
3 - x = 23 + 3x + 2\sqrt{(16 + 2x)(7 + x)}
\]

### Step 3: Isolate the square root again

Subtract \(23 + 3x\) from both sides:

\[
-20 - 4x = 2\sqrt{(16 + 2x)(7 + x)}
\]

### Step 4: Square both sides again

Square both sides to eliminate the square root:

\[
(-20 - 4x)^2 = 4[(16 + 2x)(7 + x)]
\]

### Step 5: Expand and simplify

Expand both sides:

\[
400 + 160x + 16x^2 = 4[(16)(7) + 16x + 14x + 2x^2]
\]

Simplify:

\[
400 + 160x + 16x^2 = 4[112 + 30x + 2x^2]
\]

Further simplifying:

\[
400 + 160x + 16x^2 = 448 + 120x + 8x^2
\]

### Step 6: Bring all terms to one side

\[
16x^2 - 8x^2 + 160x - 120x + 400 - 448 = 0
\]

\[
8x^2 + 40x - 48 = 0
\]

### Step 7: Solve the quadratic equation

Divide the entire equation by 8:

\[
x^2 + 5x - 6 = 0
\]

Factorize the quadratic equation:

\[
(x + 6)(x - 1) = 0
\]

So, the solutions for \(x\) are:

\[
x = -6 \quad \text{or} \quad x = 1
\]

### Step 8: Check the solutions in the original equation

1. **For \(x = -6\)**:

\[
\sqrt{3 - (-6)} - \sqrt{7 + (-6)} = \sqrt{16 + 2(-6)}
\]

Simplifies to:

\[
\sqrt{9} - \sqrt{1} = \sqrt{4}
\]

\[
3 - 1 = 2 \quad \text{(True)}
\]

2. **For \(x = 1\)**:

\[
\sqrt{3 - 1} - \sqrt{7 + 1} = \sqrt{16 + 2(1)}
\]

Simplifies to:

\[
\sqrt{2} - \sqrt{8} \neq \sqrt{18}
\]

This is **not true**.

### Final Answer:
The valid solution is \(x = -6\).

Would you like more details or have any questions? Here are some related questions for further practice:

1. How do you square both sides of an equation correctly?
2. What is the importance of checking solutions in the original equation?
3. How do you factorize a quadratic equation?
4. What are the common mistakes to avoid when solving equations involving square roots?
5. How does squaring both sides affect the solutions?

**Tip:** When dealing with square roots, always check your solutions in the original equation to avoid extraneous solutions.

Solve the equation: \(\sqrt{3 - x} - \sqrt{7 + x} = \sqrt{16 + 2x}\)

Learn how to solve the algebraic equation \(\sqrt{3 - x} - \sqrt{7 + x} = \sqrt{16 + 2x}\) step by step. This problem involves isolating square roots, squaring both sides, and solving a quadratic equation. Suitable for students in Grades 10-12.

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