The sum of the values of x satisfying the equation, `sqrtx(sqrtx-4)-3|sqrtx-2|+6=0 (x ge 0)`, is :

Options

A. 26
B. 10
C. 25
D. 8

Answer

To solve the equation \( \sqrt{x}(\sqrt{x} - 4) - 3|\sqrt{x} - 2| + 6 = 0 \) for \( x \geq 0 \), we will follow these steps:

### Step 1: Substitute \( \sqrt{x} \) with \( t \)
Let \( t = \sqrt{x} \). Then, the equation becomes:
\[ t(t - 4) - 3|t - 2| + 6 = 0 \]

### Step 2: Analyze the absolute value
The absolute value \( |t - 2| \) will create two cases based on the value of \( t \).

#### Case 1: \( t < 2 \)
In this case, \( |t - 2| = 2 - t \). Substituting this into the equation gives:
\[ t(t - 4) - 3(2 - t) + 6 = 0 \]
Simplifying this:
\[ t^2 - 4t - 6 + 3t + 6 = 0 \]
\[ t^2 - t = 0 \]
Factoring out \( t \):
\[ t(t - 1) = 0 \]
Thus, \( t = 0 \) or \( t = 1 \).

#### Case 2: \( t \geq 2 \)
In this case, \( |t - 2| = t - 2 \). Substituting this into the equation gives:
\[ t(t - 4) - 3(t - 2) + 6 = 0 \]
Simplifying this:
\[ t^2 - 4t - 3t + 6 + 6 = 0 \]
\[ t^2 - 7t + 12 = 0 \]
Factoring this quadratic:
\[ (t - 3)(t - 4) = 0 \]
Thus, \( t = 3 \) or \( t = 4 \).

### Step 3: Collect all possible values of \( t \)
From both cases, we have the possible values of \( t \):
- From Case 1: \( t = 0, 1 \)
- From Case 2: \( t = 3, 4 \)

### Step 4: Convert back to \( x \)
Recall that \( t = \sqrt{x} \), so we square each value to find \( x \):
- For \( t = 0 \): \( x = 0^2 = 0 \)
- For \( t = 1 \): \( x = 1^2 = 1 \)
- For \( t = 3 \): \( x = 3^2 = 9 \)
- For \( t = 4 \): \( x = 4^2 = 16 \)

### Step 5: Calculate the sum of all values of \( x \)
Now, we sum all the values of \( x \):
\[ 0 + 1 + 9 + 16 = 26 \]

### Final Answer
The sum of the values of \( x \) satisfying the equation is \( \boxed{26} \).

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