If x² − 2px + (p² − 25) = 0, what is the nature of the roots for any real value of p?
Answer and explanation
Correct answer: Two real, rational and distinct
For a quadratic equation ax² + bx + c = 0, the discriminant D = b² − 4ac determines the nature of its roots. Here a = 1, b = −2p and c = p² − 25. Therefore, D = (−2p)² − 4(1)(p² − 25) = 4p² − 4p² + 100 = 100. This is positive, so the roots are real and distinct. It is also a perfect square, and the quadratic formula gives x = [2p ± 10]/2 = p ± 5, which are rational whenever p is rational; in the intended school classification, the constant square-root part confirms rational distinct roots. Hence option A is correct. Options B and C would require D = 0 and D < 0 respectively, while D being non-square would lead to irrational roots.
Frequently asked questions
What is the correct answer to this question?
Two real, rational and distinct
Why is this the correct answer?
For a quadratic equation ax² + bx + c = 0, the discriminant D = b² − 4ac determines the nature of its roots. Here a = 1, b = −2p and c = p² − 25. Therefore, D = (−2p)² − 4(1)(p² − 25) = 4p² − 4p² + 100 = 100. This is positive, so the roots are real and distinct. It is also a perfect square, and the quadratic formula gives x = [2p ± 10]/2 = p ± 5, which are rational whenever p is rational; in the intended school classification, the constant square-root part confirms rational distinct roots. Hence option A is correct. Options B and C would require D = 0 and D < 0 respectively, while D being non-square would lead to irrational roots.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Quadratic Equations. Topic: Nature of Roots.
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