If `x=(sqrt(p+q)+sqrt(p-q))/(sqrt(p+q)-sqrt(p-q))` then find the value of `qx^(2)-2px+q`

Options

A. `0`
B. `1`
C. `-1`
D. `2`

Answer

To solve the problem where \( x = \frac{\sqrt{p+q} + \sqrt{p-q}}{\sqrt{p+q} - \sqrt{p-q}} \) and we need to find the value of \( qx^2 - 2px + q \), we can follow these steps:

### Step 1: Rationalize the expression for \( x \)

We start with the expression for \( x \):
\[ x = \frac{\sqrt{p+q} + \sqrt{p-q}}{\sqrt{p+q} - \sqrt{p-q}} \]

To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator:
\[ x = \frac{(\sqrt{p+q} + \sqrt{p-q})(\sqrt{p+q} + \sqrt{p-q})}{(\sqrt{p+q} - \sqrt{p-q})(\sqrt{p+q} + \sqrt{p-q})} \]

### Step 2: Simplify the denominator

The denominator simplifies using the difference of squares:
\[ (\sqrt{p+q})^2 - (\sqrt{p-q})^2 = (p+q) - (p-q) = 2q \]

### Step 3: Simplify the numerator

The numerator becomes:
\[ (\sqrt{p+q} + \sqrt{p-q})^2 = (\sqrt{p+q})^2 + 2\sqrt{(p+q)(p-q)} + (\sqrt{p-q})^2 = (p+q) + (p-q) + 2\sqrt{(p+q)(p-q)} \]
This simplifies to:
\[ 2p + 2\sqrt{(p+q)(p-q)} = 2p + 2\sqrt{p^2 - q^2} \]

### Step 4: Combine the results for \( x \)

Now substituting back, we have:
\[ x = \frac{2p + 2\sqrt{p^2 - q^2}}{2q} = \frac{p + \sqrt{p^2 - q^2}}{q} \]

### Step 5: Calculate \( x^2 \)

Next, we calculate \( x^2 \):
\[ x^2 = \left(\frac{p + \sqrt{p^2 - q^2}}{q}\right)^2 = \frac{(p + \sqrt{p^2 - q^2})^2}{q^2} \]
Expanding the numerator:
\[ (p + \sqrt{p^2 - q^2})^2 = p^2 + 2p\sqrt{p^2 - q^2} + (p^2 - q^2) = 2p^2 - q^2 + 2p\sqrt{p^2 - q^2} \]

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

### Step 6: Substitute \( x \) and \( x^2 \) into \( qx^2 - 2px + q \)

Now we substitute \( x^2 \) and \( x \) into the expression \( qx^2 - 2px + q \):
\[ qx^2 = q \cdot \frac{2p^2 - q^2 + 2p\sqrt{p^2 - q^2}}{q^2} = \frac{q(2p^2 - q^2 + 2p\sqrt{p^2 - q^2})}{q^2} \]
This simplifies to:
\[ \frac{2p^2 - q^2 + 2p\sqrt{p^2 - q^2}}{q} \]

Now substituting \( -2px \):
\[ -2px = -2p \cdot \frac{p + \sqrt{p^2 - q^2}}{q} = \frac{-2p^2 - 2p\sqrt{p^2 - q^2}}{q} \]

Combining these:
\[ qx^2 - 2px + q = \frac{2p^2 - q^2 + 2p\sqrt{p^2 - q^2} - 2p^2 - 2p\sqrt{p^2 - q^2} + q}{q} \]
This simplifies to:
\[ \frac{-q^2 + q}{q} = \frac{q - q^2}{q} = 0 \]

### Final Result

Thus, the value of \( qx^2 - 2px + q \) is:
\[ \boxed{0} \]