https://file.aiquickdraw.com/mathbot/file/1725385884782wto27jj0.jpg

The equation in the image is:

\[
\sqrt{x + 1} + \sqrt{x + 6} = 5
\]

To solve this equation, we can follow these steps:

### Step 1: Isolate one of the square roots
Let's isolate \(\sqrt{x + 1}\) on one side of the equation:

\[
\sqrt{x + 1} = 5 - \sqrt{x + 6}
\]

### Step 2: Square both sides of the equation
To eliminate the square root on the left side, square both sides:

\[
(\sqrt{x + 1})^2 = (5 - \sqrt{x + 6})^2
\]

This simplifies to:

\[
x + 1 = 25 - 10\sqrt{x + 6} + (x + 6)
\]

Simplify further:

\[
x + 1 = x + 31 - 10\sqrt{x + 6}
\]

### Step 3: Isolate the square root term
Now, subtract \(x + 31\) from both sides:

\[
x + 1 - (x + 31) = -10\sqrt{x + 6}
\]

Simplify:

\[
-30 = -10\sqrt{x + 6}
\]

Divide by -10:

\[
3 = \sqrt{x + 6}
\]

### Step 4: Square both sides again
Square both sides to remove the square root:

\[
9 = x + 6
\]

### Step 5: Solve for \(x\)
Subtract 6 from both sides:

\[
x = 3
\]

### Step 6: Verify the solution
Substitute \(x = 3\) back into the original equation to verify:

\[
\sqrt{3 + 1} + \sqrt{3 + 6} = \sqrt{4} + \sqrt{9} = 2 + 3 = 5
\]

Since both sides are equal, the solution is correct.

### Final Answer:
The solution to the equation is \(x = 3\).

Would you like any more details or have any further questions?

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Here are 5 related questions to explore:

1. How do you solve similar equations involving square roots?
2. What happens if the square root equation has no real solution?
3. Can you solve an equation involving cube roots similarly?
4. How would you solve an equation with two different radicals?
5. What are the implications if both radicals cannot be isolated easily?

**Tip:** Always check for extraneous solutions when solving square root equations by substituting the found solutions back into the original equation.

Solve the equation: \(\sqrt{x + 1} + \sqrt{x + 6} = 5\)

Learn how to solve the radical equation involving square roots: sqrt(x + 1) + sqrt(x + 6) = 5. This step-by-step solution covers isolating radicals, squaring both sides, and verifying the solution. Suitable for high school students in Grades 10-12.

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