How many of the following values of x would satisfy the equation qx^{2}-2px+q=0 ?<br><br>1. x=\frac{\sqrt{p+q}+\sqrt{p-q}}{\sqrt{p+q}-\sqrt{p-q}} where p \gt q<br>2. x=\frac{p+\sqrt{p^{2}-q^{2}}}{q} where p \gt q<br>3. x=\frac{\sqrt{p+q}}{\sqrt{p-q}} where p \gt q<br><br>Select the correct answer using the code given below:
- A. Only one value
- B. Only two values ✓
- C. All three values
- D. None
Correct Answer: B. Only two values
Explanation
The roots of qx^2 - 2px + q = 0 are x = \frac{2p \pm \sqrt{4p^2-4q^2}}{2q} = \frac{p \pm \sqrt{p^2-q^2}}{q}. Expression 2 perfectly matches one root. Rationalizing the denominator of Expression 1 yields \frac{p+\sqrt{p^2-q^2}}{q}, which is the identical root. Expression 3 does not solve the equation. Thus, two expressions provide valid values.
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