If x² − 2(k − 3)x + (k² − 8k + 20) = 0 has no real roots, what is the correct condition on k?
Answer and explanation
Correct answer: k < 11/2
For a quadratic equation ax² + bx + c = 0, the roots are non-real when the discriminant D = b² − 4ac is negative. Here a = 1, b = −2(k − 3), and c = k² − 8k + 20. Hence D = [−2(k − 3)]² − 4(k² − 8k + 20) = 4(k − 3)² − 4(k² − 8k + 20) = 4(2k − 11). The condition D < 0 gives 4(2k − 11) < 0, so 2k − 11 < 0 and therefore k < 11/2. Equality gives repeated real roots, while larger values give two distinct real roots.
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What is the correct answer to this question?
k < 11/2
Why is this the correct answer?
For a quadratic equation ax² + bx + c = 0, the roots are non-real when the discriminant D = b² − 4ac is negative. Here a = 1, b = −2(k − 3), and c = k² − 8k + 20. Hence D = [−2(k − 3)]² − 4(k² − 8k + 20) = 4(k − 3)² − 4(k² − 8k + 20) = 4(2k − 11). The condition D < 0 gives 4(2k − 11) < 0, so 2k − 11 < 0 and therefore k < 11/2. Equality gives repeated real roots, while larger values give two distinct real 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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