Question:
If the roots of the quadratic equation $x^2 - 2px + q^2 = 0$ are real and distinct, then:
If the roots of the quadratic equation $x^2 - 2px + q^2 = 0$ are real and distinct, then:
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"Real and distinct" strictly translates to a strict inequality ($>$). This immediately rules out options (C) and (D) containing $\ge$ and $\le$.
Updated On: Jun 3, 2026
- $|p| > |q|$
- $|p| < |q|$
- $p^2 \ge q^2$
- $p^2 \le q^2$
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The Correct Option is A
Solution and Explanation
Step 1: Concept
For any quadratic equation $ax^2 + bx + c = 0$ to have real and distinct roots, its discriminant $D = b^2 - 4ac$ must be strictly greater than zero ($D > 0$).
Step 2: Meaning
For the equation $x^2 - 2px + q^2 = 0$, the coefficients are $a = 1$, $b = -2p$, and $c = q^2$.
Step 3: Analysis
Calculate the discriminant: \[ D = (-2p)^2 - 4(1)(q^2) = 4p^2 - 4q^2 \] Since the roots are real and distinct: \[ D > 0 \implies 4p^2 - 4q^2 > 0 \implies p^2 > q^2 \] Taking the square root on both sides: \[ |p| > |q| \]
Step 4: Conclusion
Thus, the condition for real and distinct roots is $|p| > |q|$.
Final Answer: (A)
For any quadratic equation $ax^2 + bx + c = 0$ to have real and distinct roots, its discriminant $D = b^2 - 4ac$ must be strictly greater than zero ($D > 0$).
Step 2: Meaning
For the equation $x^2 - 2px + q^2 = 0$, the coefficients are $a = 1$, $b = -2p$, and $c = q^2$.
Step 3: Analysis
Calculate the discriminant: \[ D = (-2p)^2 - 4(1)(q^2) = 4p^2 - 4q^2 \] Since the roots are real and distinct: \[ D > 0 \implies 4p^2 - 4q^2 > 0 \implies p^2 > q^2 \] Taking the square root on both sides: \[ |p| > |q| \]
Step 4: Conclusion
Thus, the condition for real and distinct roots is $|p| > |q|$.
Final Answer: (A)
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