A student writes D=16 for x²−2(k−2)x+k²=0. What is the correct discriminant D?
Answer and explanation
Correct answer: 16(1−k)
For a quadratic equation ax² + bx + c = 0, the discriminant is D = b² − 4ac. Compare x² − 2(k−2)x + k² = 0 with the standard form. Here a = 1, b = −2(k−2), and c = k². Therefore D = [−2(k−2)]² − 4(1)(k²) = 4(k−2)² − 4k². Expanding gives 4(k² − 4k + 4) − 4k² = 4k² − 16k + 16 − 4k² = 16 − 16k = 16(1−k). Hence option A is correct. Option B results from incorrectly discarding the k-dependent terms. Option C has an incorrect sign and expression, while option D is not obtained from b² − 4ac. Because k is a parameter, the correct discriminant remains an algebraic expression in k.
Frequently asked questions
What is the correct answer to this question?
16(1−k)
Why is this the correct answer?
For a quadratic equation ax² + bx + c = 0, the discriminant is D = b² − 4ac. Compare x² − 2(k−2)x + k² = 0 with the standard form. Here a = 1, b = −2(k−2), and c = k². Therefore D = [−2(k−2)]² − 4(1)(k²) = 4(k−2)² − 4k². Expanding gives 4(k² − 4k + 4) − 4k² = 4k² − 16k + 16 − 4k² = 16 − 16k = 16(1−k). Hence option A is correct. Option B results from incorrectly discarding the k-dependent terms. Option C has an incorrect sign and expression, while option D is not obtained from b² − 4ac. Because k is a parameter, the correct discriminant remains an algebraic expression in k.
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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