If ` x = ( sqrt( p + 2 q)+ sqrt(p - 2q))/( sqrt( p + 2 q) - sqrt( p - 2q))` then ` q x^(2) - px + q ` equals to

Options

A. 1
B. 0
C. 2
D. 3

Answer

To solve the problem, we start with the expression given for \( x \):

\[ x = \frac{\sqrt{p + 2q} + \sqrt{p - 2q}}{\sqrt{p + 2q} - \sqrt{p - 2q}} \]

We need to find the value of \( qx^2 - px + q \).

### Step 1: Apply the Conjugate

To simplify \( x \), we can multiply the numerator and denominator by the conjugate of the denominator:

\[ x = \frac{(\sqrt{p + 2q} + \sqrt{p - 2q})(\sqrt{p + 2q} + \sqrt{p - 2q})}{(\sqrt{p + 2q} - \sqrt{p - 2q})(\sqrt{p + 2q} + \sqrt{p - 2q})} \]

### Step 2: Simplify the Denominator

The denominator simplifies using the difference of squares:

\[ (\sqrt{p + 2q})^2 - (\sqrt{p - 2q})^2 = (p + 2q) - (p - 2q) = 4q \]

### Step 3: Simplify the Numerator

The numerator simplifies as follows:

\[ (\sqrt{p + 2q} + \sqrt{p - 2q})^2 = (p + 2q) + (p - 2q) + 2\sqrt{(p + 2q)(p - 2q)} = 2p + 2\sqrt{(p + 2q)(p - 2q)} \]

### Step 4: Combine the Results

Now we can write \( x \):

\[ x = \frac{2p + 2\sqrt{(p + 2q)(p - 2q)}}{4q} = \frac{p + \sqrt{(p + 2q)(p - 2q)}}{2q} \]

### Step 5: Square \( x \)

Next, we find \( x^2 \):

\[ x^2 = \left(\frac{p + \sqrt{(p + 2q)(p - 2q)}}{2q}\right)^2 = \frac{(p + \sqrt{(p + 2q)(p - 2q)})^2}{4q^2} \]

### Step 6: Substitute into the Expression

Now we substitute \( x^2 \) into \( qx^2 - px + q \):

\[ qx^2 = q \cdot \frac{(p + \sqrt{(p + 2q)(p - 2q)})^2}{4q^2} = \frac{(p + \sqrt{(p + 2q)(p - 2q)})^2}{4q} \]

### Step 7: Combine All Terms

Now we need to combine \( qx^2 - px + q \):

\[ qx^2 - px + q = \frac{(p + \sqrt{(p + 2q)(p - 2q)})^2}{4q} - p\left(\frac{p + \sqrt{(p + 2q)(p - 2q)}}{2q}\right) + q \]

### Step 8: Simplify the Expression

After simplification, we find that all terms cancel out, leading to:

\[ qx^2 - px + q = 0 \]

### Final Answer

Thus, the value of \( qx^2 - px + q \) is:

\[ \boxed{0} \]