If x + sqrt(x-2)  = 8 then what is the sum of all values of x ?

To solve the equation \( x + \sqrt{x-2} = 8 \) and find the sum of all values of \( x \), let's proceed step-by-step:

### Step 1: Isolate the square root term

Given the equation:

\[
x + \sqrt{x-2} = 8
\]

Subtract \( x \) from both sides to isolate the square root:

\[
\sqrt{x-2} = 8 - x
\]

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

\[
(\sqrt{x-2})^2 = (8 - x)^2
\]

\[
x-2 = (8-x)^2
\]

### Step 3: Expand and simplify the equation

Expand \( (8-x)^2 \):

\[
x-2 = (8-x)(8-x)
\]

\[
x-2 = 64 - 16x + x^2
\]

### Step 4: Rearrange into a standard quadratic form

Bring all terms to one side:

\[
x^2 - 16x + 64 - x + 2 = 0
\]

\[
x^2 - 17x + 66 = 0
\]

### Step 5: Solve the quadratic equation

Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) with \( a = 1 \), \( b = -17 \), and \( c = 66 \):

\[
x = \frac{-(-17) \pm \sqrt{(-17)^2 - 4 \cdot 1 \cdot 66}}{2 \cdot 1}
\]

\[
x = \frac{17 \pm \sqrt{289 - 264}}{2}
\]

\[
x = \frac{17 \pm \sqrt{25}}{2}
\]

\[
x = \frac{17 \pm 5}{2}
\]

This gives two solutions:

\[
x = \frac{22}{2} = 11 \quad \text{and} \quad x = \frac{12}{2} = 6
\]

### Step 6: Verify the solutions

1. **For \( x = 11 \)**:

   \[
   x + \sqrt{x-2} = 11 + \sqrt{11-2} = 11 + 3 = 14 \quad (\text{which is not } 8)
   \]

   Hence, \( x = 11 \) is **not** a solution.

2. **For \( x = 6 \)**:

   \[
   x + \sqrt{x-2} = 6 + \sqrt{6-2} = 6 + 2 = 8 \quad (\text{which is } 8)
   \]

   Hence, \( x = 6 \) is a valid solution.

### Step 7: Sum of all valid solutions

Since the only valid solution is \( x = 6 \), the sum of all values of \( x \) is:

\[
\text{Sum} = 6
\]

Therefore, the sum of all values of \( x \) is **6**.

Would you like more details, or do you have any questions?

### Related Questions

1. How would you solve \( x + \sqrt{x-3} = 9 \)?
2. What are the conditions for a square root equation to have real solutions?
3. If \( x + \sqrt{x-1} = 10 \), what are the possible values of \( x \)?
4. Can you generalize a method for solving equations of the form \( x + \sqrt{x-a} = b \)?
5. What happens if you solve \( x + \sqrt{x-2} = 7 \)?

### Tip
Always verify potential solutions in the original equation, especially when dealing with square roots, as squaring can introduce extraneous solutions.

Learn how to solve the equation x + sqrt(x-2) = 8 with a detailed step-by-step solution. This problem involves algebraic manipulation, square roots, and the quadratic formula. Suitable for high school students in Grades 9-12.

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