If \(k\) is a real number, what is the correct condition for the equation \(x^2+2(k-1)x+(k+5)=0\) to have no real roots?
Answer and explanation
Correct answer: \(-1<k<4\)
A quadratic equation has no real roots when its discriminant satisfies \(D<0\). Here, \(a=1\), \(b=2(k-1)\), and \(c=k+5\). Hence, \(D=[2(k-1)]^2-4(k+5)=4(k^2-3k-4)=4(k-4)(k+1)\). Therefore, \((k-4)(k+1)<0\), which holds for \(-1<k<4\). At \(k=-1\) or \(k=4\), \(D=0\), so the roots are equal and real, not absent. Exam tip: for questions on the nature of roots, first calculate \(D\) and then analyse its sign.
Frequently asked questions
What is the correct answer to this question?
\(-1<k<4\)
Why is this the correct answer?
A quadratic equation has no real roots when its discriminant satisfies \(D<0\). Here, \(a=1\), \(b=2(k-1)\), and \(c=k+5\). Hence, \(D=[2(k-1)]^2-4(k+5)=4(k^2-3k-4)=4(k-4)(k+1)\). Therefore, \((k-4)(k+1)<0\), which holds for \(-1<k<4\). At \(k=-1\) or \(k=4\), \(D=0\), so the roots are equal and real, not absent. Exam tip: for questions on the nature of roots, first calculate \(D\) and then analyse its sign.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Quadratic Equations. Topic: Nature of Roots.
Student feedback
Was this question useful?
👍 0 Helpful 👎 0 Not helpful
Yes 0% No 0%
0 responsesStudent Reviews
No published reviews yet.
