We are given the equation:

\[
\sqrt{x - 1} = 7 - x
\]

To solve this equation, follow these steps:

### Step 1: Square both sides
Squaring both sides will eliminate the square root. The equation becomes:

\[
(\sqrt{x - 1})^2 = (7 - x)^2
\]

This simplifies to:

\[
x - 1 = (7 - x)^2
\]

### Step 2: Expand the right-hand side
Now, expand \((7 - x)^2\):

\[
(7 - x)^2 = 49 - 14x + x^2
\]

So the equation becomes:

\[
x - 1 = x^2 - 14x + 49
\]

### Step 3: Rearrange the equation
Move all terms to one side of the equation:

\[
0 = x^2 - 14x + 49 - (x - 1)
\]

Simplify this:

\[
0 = x^2 - 14x + 49 - x + 1
\]

\[
0 = x^2 - 15x + 50
\]

### Step 4: Solve the quadratic equation
We now solve the quadratic equation:

\[
x^2 - 15x + 50 = 0
\]

Use the quadratic formula:

\[
x = \frac{-(-15) \pm \sqrt{(-15)^2 - 4(1)(50)}}{2(1)}
\]

Simplifying:

\[
x = \frac{15 \pm \sqrt{225 - 200}}{2}
\]

\[
x = \frac{15 \pm \sqrt{25}}{2}
\]

\[
x = \frac{15 \pm 5}{2}
\]

This gives two solutions:

\[
x_1 = \frac{15 + 5}{2} = 10 \quad \text{and} \quad x_2 = \frac{15 - 5}{2} = 5
\]

### Step 5: Check for extraneous solutions
We must check if both solutions satisfy the original equation.

- For \(x = 10\):
  \[
  \sqrt{10 - 1} = 7 - 10
  \]
  \[
  \sqrt{9} = -3 \quad (\text{false, as } \sqrt{9} = 3)
  \]
  So, \(x = 10\) is not a solution.

- For \(x = 5\):
  \[
  \sqrt{5 - 1} = 7 - 5
  \]
  \[
  \sqrt{4} = 2 \quad (\text{true, as } 2 = 2)
  \]
  So, \(x = 5\) is a valid solution.

### Final Answer:
The solution is \(x = 5\).

Would you like more details on any step? Here are some related questions:

1. How do you expand a binomial like \((7 - x)^2\)?
2. Can you explain more about squaring both sides in equations involving square roots?
3. How does the quadratic formula work in more complex equations?
4. What are extraneous solutions and why do they occur?
5. How do you check solutions in radical equations?

**Tip**: Always check your solutions when working with square root equations to avoid extraneous results.

Learn how to solve the radical equation √(x - 1) = 7 - x using squaring both sides and solving the resulting quadratic equation. Understand key algebraic concepts, including the quadratic formula and checking for extraneous solutions. Suitable for students in Grades 9-10.

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