If the roots of the equation \(x^2-2(k+2)x+(k^2+3k+7)=0\) are real, which condition on \(k\) is necessary?
Answer and explanation
Correct answer: \(k\geq 3\)
For a quadratic equation to have real roots, its discriminant must satisfy \(D\geq 0\). Here, \(D=[-2(k+2)]^2-4(k^2+3k+7)=4(k-3)\). Therefore, \(4(k-3)\geq 0\), which gives \(k\geq 3\). Option B is incorrect because \(k=3\) also gives real, equal roots. Exam tip: For questions about real roots, begin by applying \(D\geq 0\).
Frequently asked questions
What is the correct answer to this question?
\(k\geq 3\)
Why is this the correct answer?
For a quadratic equation to have real roots, its discriminant must satisfy \(D\geq 0\). Here, \(D=[-2(k+2)]^2-4(k^2+3k+7)=4(k-3)\). Therefore, \(4(k-3)\geq 0\), which gives \(k\geq 3\). Option B is incorrect because \(k=3\) also gives real, equal roots. Exam tip: For questions about real roots, begin by applying \(D\geq 0\).
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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