https://file.aiquickdraw.com/mathbot/file/1725459159138lwdm74b0.jpg

The equation provided in the image is:

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

To solve this equation, we'll proceed step by step.

### Step 1: Isolate one of the square roots
First, let's try to isolate one of the square roots. We can start by isolating \(\sqrt{16 + 2x}\):

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

Add \(\sqrt{7 + x}\) to both sides:

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

### Step 2: Square both sides
Next, square both sides of the equation to eliminate the square roots:

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

On the left side:

\[
3 - x
\]

On the right side, use the identity \((a + b)^2 = a^2 + 2ab + b^2\):

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

Simplifying:

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

### Step 3: Isolate the square root again
Now, isolate the square root term:

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

Simplify:

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

Divide by 2:

\[
-2x - 10 = \sqrt{(7 + x)(16 + 2x)}
\]

### Step 4: Square both sides again
Square both sides again:

\[
(-2x - 10)^2 = (7 + x)(16 + 2x)
\]

Expand both sides:

\[
4x^2 + 40x + 100 = 112 + 30x + 2x^2
\]

### Step 5: Simplify and solve the quadratic equation
Move all terms to one side:

\[
4x^2 + 40x + 100 - 112 - 30x - 2x^2 = 0
\]

Simplify:

\[
2x^2 + 10x - 12 = 0
\]

Divide by 2:

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

Factor the quadratic:

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

So, the solutions are:

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

### Step 6: Verify the solutions
We need to check both potential solutions in the original equation.

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

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

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

\[
3 - 1 = 2 \quad \text{which is true}
\]

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

\[
\sqrt{3 - 

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

Learn how to solve the square root equation \(\sqrt{3 - x} - \sqrt{7 + x} = \sqrt{16 + 2x}\) with a detailed step-by-step solution. This problem covers algebraic concepts, including square roots and quadratic equations, and is suitable for students in Grades 10-12.

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