https://file.aiquickdraw.com/mathbot/file/1727034898395h5hqqshq.jpg

The equation in the image is:

\[
\sqrt{x + 6} = 6 - 2\sqrt{5 - x}
\]

To solve this equation, we need to eliminate the square roots by squaring both sides. Let's go step by step.

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

\[
\left(\sqrt{x + 6}\right)^2 = \left(6 - 2\sqrt{5 - x}\right)^2
\]

This simplifies to:

\[
x + 6 = (6 - 2\sqrt{5 - x})^2
\]

### Step 2: Expand the right-hand side
Use the formula \((a - b)^2 = a^2 - 2ab + b^2\) for the right side, where \(a = 6\) and \(b = 2\sqrt{5 - x}\):

\[
(6 - 2\sqrt{5 - x})^2 = 6^2 - 2(6)(2\sqrt{5 - x}) + (2\sqrt{5 - x})^2
\]

This simplifies to:

\[
36 - 24\sqrt{5 - x} + 4(5 - x)
\]

Expanding the last term:

\[
36 - 24\sqrt{5 - x} + 20 - 4x
\]

Now, combine like terms:

\[
56 - 4x - 24\sqrt{5 - x}
\]

### Step 3: Set the equation up for further solving
Now we have:

\[
x + 6 = 56 - 4x - 24\sqrt{5 - x}
\]

Let's move all the non-square root terms to one side:

\[
x + 4x + 6 - 56 = -24\sqrt{5 - x}
\]

This simplifies to:

\[
5x - 50 = -24\sqrt{5 - x}
\]

### Step 4: Isolate the square root term
Solve for the square root:

\[
\frac{5x - 50}{-24} = \sqrt{5 - x}
\]

Simplify:

\[
\frac{50 - 5x}{24} = \sqrt{5 - x}
\]

### Step 5: Square both sides again
Square both sides to eliminate the square root:

\[
\left(\frac{50 - 5x}{24}\right)^2 = 5 - x
\]

Now we can expand and solve this equation for \(x\).

I will proceed with the calculation.The solutions to the equation are:

\[
x = \frac{-38}{25} + \frac{24\sqrt{19}}{25} \quad \text{or} \quad x = \frac{-38}{25} - \frac{24\sqrt{19}}{25}
\]

These are the two possible solutions for \(x\). Since we are dealing with square roots, we must check for extraneous solutions by substituting these values back into the original equation. Would you like me to verify these solutions or provide further details?

Here are some related questions you might find helpful:
1. What are extraneous solutions, and why do they occur in square root equations?
2. How do you simplify complex square root equations?
3. What is the geometric interpretation of solving square root equations?
4. How do squaring both sides of an equation affect potential solutions?
5. Can this type of problem be solved using graphical methods?

**Tip**: Always check for extraneous solutions when solving equations involving square roots or squaring both sides.

Solve the equation \(\sqrt{x + 6} = 6 - 2\sqrt{5 - x}\).

Learn how to solve the square root equation \(\sqrt{x + 6} = 6 - 2\sqrt{5 - x}\) step-by-step, covering key algebraic techniques such as squaring both sides, expanding binomials, and solving radical equations. Suitable for students in Grades 10-12.

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