Which condition on \(k\) ensures that the equation \(x^2+2x+k=0\) has no real roots?
Answer and explanation
Correct answer: \(k>1\)
A quadratic equation has no real roots when its discriminant satisfies \(D<0\). Here, \(D=b^2-4ac=2^2-4(1)(k)=4-4k\). Thus, \(4-4k<0\), which gives \(k>1\). If \(k=1\), then \(D=0\), so the equation has two equal real roots; hence option B is incorrect. Exam tip: Check the sign of the discriminant first to determine the nature of the roots.
Frequently asked questions
What is the correct answer to this question?
\(k>1\)
Why is this the correct answer?
A quadratic equation has no real roots when its discriminant satisfies \(D<0\). Here, \(D=b^2-4ac=2^2-4(1)(k)=4-4k\). Thus, \(4-4k<0\), which gives \(k>1\). If \(k=1\), then \(D=0\), so the equation has two equal real roots; hence option B is incorrect. Exam tip: Check the sign of the discriminant first to determine the nature of the 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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